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G = C24.6A4  order 192 = 26·3

6th non-split extension by C24 of A4 acting faithfully

metabelian, soluble, monomial

Aliases: C24.6A4, C4⋊1D4⋊C6, C42⋊(C2×C6), C42⋊C6⋊C2, C42⋊2C2⋊C6, C42⋊C3⋊2C22, C23.5(C2×A4), C23.A4⋊1C2, C22.54C24⋊C3, C22.5(C22×A4), SmallGroup(192,1008)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C42 — C24.6A4
Chief series
C1 — C22 — C42 — C42⋊C3 — C23.A4 — C24.6A4
Lower central
C42 — C24.6A4
Upper central
C1

Generators and relations for C24.6A4
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g3=1, e2=dc=gcg-1=cd, f2=gdg-1=c, ab=ba, faf-1=ac=ca, ad=da, eae-1=acd, ag=ga, ebe-1=bc=cb, fbf-1=bd=db, bg=gb, ce=ec, cf=fc, gfg-1=de=ed, df=fd, ef=fe, geg-1=cef >

Subgroups: 342 in 67 conjugacy classes, 16 normal (10 characteristic)
C1, C2, C3, C4, C22, C22, C6, C2×C4, D4, C23, C23, C23, A4, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C24, C2×A4, C22≀C2, C4⋊D4, C22.D4, C42⋊2C2, C4⋊1D4, C42⋊C3, C22×A4, C22.54C24, C42⋊C6, C23.A4, C24.6A4
Quotients: C1, C2, C3, C22, C6, A4, C2×C6, C2×A4, C22×A4, C24.6A4

Character table of C24.6A4

 class 12A2B2C2D2E3A3B4A4B4C6A6B6C6D6E6F
 size 13444121616121212161616161616
ρ111111111111111111    trivial
ρ211-1-11-1111-111-1-1-11-1    linear of order 2
ρ3111-1-1-11111-1-1-111-1-1    linear of order 2
ρ411-11-11111-1-1-11-1-1-11    linear of order 2
ρ5111-1-1-1ζ32ζ311-1ζ65ζ65ζ32ζ3ζ6ζ6    linear of order 6
ρ611-1-11-1ζ3ζ321-11ζ32ζ6ζ65ζ6ζ3ζ65    linear of order 6
ρ7111111ζ3ζ32111ζ32ζ32ζ3ζ32ζ3ζ3    linear of order 3
ρ811-11-11ζ3ζ321-1-1ζ6ζ32ζ65ζ6ζ65ζ3    linear of order 6
ρ9111-1-1-1ζ3ζ3211-1ζ6ζ6ζ3ζ32ζ65ζ65    linear of order 6
ρ1011-1-11-1ζ32ζ31-11ζ3ζ65ζ6ζ65ζ32ζ6    linear of order 6
ρ1111-11-11ζ32ζ31-1-1ζ65ζ3ζ6ζ65ζ6ζ32    linear of order 6
ρ12111111ζ32ζ3111ζ3ζ3ζ32ζ3ζ32ζ32    linear of order 3
ρ1333-3-33100-11-1000000    orthogonal lifted from C2×A4
ρ1433-33-3-100-111000000    orthogonal lifted from C2×A4
ρ15333-3-3100-1-11000000    orthogonal lifted from C2×A4
ρ1633333-100-1-1-1000000    orthogonal lifted from A4
ρ1712-4000000000000000    orthogonal faithful

Permutation representations of C24.6A4
►On 16 points - transitive group 16T420
Generators in S16
(2 4)(5 12)(6 11)(7 10)(8 9)(14 16)

(2 14)(4 16)(5 10)(6 8)(7 12)(9 11)

(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)

(1 15)(2 16)(3 13)(4 14)(5 10)(6 11)(7 12)(8 9)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 12 13 5)(2 9 14 6)(3 10 15 7)(4 11 16 8)

(2 10 6)(3 13 15)(4 7 11)(5 8 14)(9 16 12)
 
G:=sub<Sym(16)| (2,4)(5,12)(6,11)(7,10)(8,9)(14,16), (2,14)(4,16)(5,10)(6,8)(7,12)(9,11), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12,13,5)(2,9,14,6)(3,10,15,7)(4,11,16,8), (2,10,6)(3,13,15)(4,7,11)(5,8,14)(9,16,12)>;
 
G:=Group( (2,4)(5,12)(6,11)(7,10)(8,9)(14,16), (2,14)(4,16)(5,10)(6,8)(7,12)(9,11), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12,13,5)(2,9,14,6)(3,10,15,7)(4,11,16,8), (2,10,6)(3,13,15)(4,7,11)(5,8,14)(9,16,12) );
 
G=PermutationGroup([[(2,4),(5,12),(6,11),(7,10),(8,9),(14,16)], [(2,14),(4,16),(5,10),(6,8),(7,12),(9,11)], [(1,13),(2,14),(3,15),(4,16),(5,12),(6,9),(7,10),(8,11)], [(1,15),(2,16),(3,13),(4,14),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,12,13,5),(2,9,14,6),(3,10,15,7),(4,11,16,8)], [(2,10,6),(3,13,15),(4,7,11),(5,8,14),(9,16,12)]])
 
G:=TransitiveGroup(16,420);
 

►On 24 points - transitive group 24T368
Generators in S24
(1 5)(2 6)(10 12)(14 16)(17 19)(21 23)

(2 6)(3 8)(9 11)(10 12)(13 15)(17 19)

(1 5)(2 6)(3 8)(4 7)(13 15)(14 16)(17 19)(18 20)

(1 5)(2 6)(3 8)(4 7)(9 11)(10 12)(21 23)(22 24)

(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 3 5 8)(2 4 6 7)(9 22)(10 23)(11 24)(12 21)(13 17 15 19)(14 18 16 20)

(1 21 16)(2 10 19)(3 9 13)(4 24 20)(5 23 14)(6 12 17)(7 22 18)(8 11 15)
 
G:=sub<Sym(24)| (1,5)(2,6)(10,12)(14,16)(17,19)(21,23), (2,6)(3,8)(9,11)(10,12)(13,15)(17,19), (1,5)(2,6)(3,8)(4,7)(13,15)(14,16)(17,19)(18,20), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3,5,8)(2,4,6,7)(9,22)(10,23)(11,24)(12,21)(13,17,15,19)(14,18,16,20), (1,21,16)(2,10,19)(3,9,13)(4,24,20)(5,23,14)(6,12,17)(7,22,18)(8,11,15)>;
 
G:=Group( (1,5)(2,6)(10,12)(14,16)(17,19)(21,23), (2,6)(3,8)(9,11)(10,12)(13,15)(17,19), (1,5)(2,6)(3,8)(4,7)(13,15)(14,16)(17,19)(18,20), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3,5,8)(2,4,6,7)(9,22)(10,23)(11,24)(12,21)(13,17,15,19)(14,18,16,20), (1,21,16)(2,10,19)(3,9,13)(4,24,20)(5,23,14)(6,12,17)(7,22,18)(8,11,15) );
 
G=PermutationGroup([[(1,5),(2,6),(10,12),(14,16),(17,19),(21,23)], [(2,6),(3,8),(9,11),(10,12),(13,15),(17,19)], [(1,5),(2,6),(3,8),(4,7),(13,15),(14,16),(17,19),(18,20)], [(1,5),(2,6),(3,8),(4,7),(9,11),(10,12),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,3,5,8),(2,4,6,7),(9,22),(10,23),(11,24),(12,21),(13,17,15,19),(14,18,16,20)], [(1,21,16),(2,10,19),(3,9,13),(4,24,20),(5,23,14),(6,12,17),(7,22,18),(8,11,15)]])
 
G:=TransitiveGroup(24,368);
 

►On 24 points - transitive group 24T373
Generators in S24
(1 4)(2 3)(5 7)(6 8)(9 18)(10 17)(11 20)(12 19)(13 16)(14 15)(21 24)(22 23)

(2 7)(3 5)(10 12)(13 15)(14 16)(17 19)

(1 8)(2 7)(3 5)(4 6)(9 11)(10 12)(17 19)(18 20)

(1 8)(2 7)(3 5)(4 6)(13 15)(14 16)(21 23)(22 24)

(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 5 8 3)(2 6 7 4)(9 20 11 18)(10 17 12 19)(13 23)(14 24)(15 21)(16 22)

(1 24 11)(2 16 17)(3 13 10)(4 21 20)(5 15 12)(6 23 18)(7 14 19)(8 22 9)
 
G:=sub<Sym(24)| (1,4)(2,3)(5,7)(6,8)(9,18)(10,17)(11,20)(12,19)(13,16)(14,15)(21,24)(22,23), (2,7)(3,5)(10,12)(13,15)(14,16)(17,19), (1,8)(2,7)(3,5)(4,6)(9,11)(10,12)(17,19)(18,20), (1,8)(2,7)(3,5)(4,6)(13,15)(14,16)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,5,8,3)(2,6,7,4)(9,20,11,18)(10,17,12,19)(13,23)(14,24)(15,21)(16,22), (1,24,11)(2,16,17)(3,13,10)(4,21,20)(5,15,12)(6,23,18)(7,14,19)(8,22,9)>;
 
G:=Group( (1,4)(2,3)(5,7)(6,8)(9,18)(10,17)(11,20)(12,19)(13,16)(14,15)(21,24)(22,23), (2,7)(3,5)(10,12)(13,15)(14,16)(17,19), (1,8)(2,7)(3,5)(4,6)(9,11)(10,12)(17,19)(18,20), (1,8)(2,7)(3,5)(4,6)(13,15)(14,16)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,5,8,3)(2,6,7,4)(9,20,11,18)(10,17,12,19)(13,23)(14,24)(15,21)(16,22), (1,24,11)(2,16,17)(3,13,10)(4,21,20)(5,15,12)(6,23,18)(7,14,19)(8,22,9) );
 
G=PermutationGroup([[(1,4),(2,3),(5,7),(6,8),(9,18),(10,17),(11,20),(12,19),(13,16),(14,15),(21,24),(22,23)], [(2,7),(3,5),(10,12),(13,15),(14,16),(17,19)], [(1,8),(2,7),(3,5),(4,6),(9,11),(10,12),(17,19),(18,20)], [(1,8),(2,7),(3,5),(4,6),(13,15),(14,16),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,5,8,3),(2,6,7,4),(9,20,11,18),(10,17,12,19),(13,23),(14,24),(15,21),(16,22)], [(1,24,11),(2,16,17),(3,13,10),(4,21,20),(5,15,12),(6,23,18),(7,14,19),(8,22,9)]])
 
G:=TransitiveGroup(24,373);
 

►On 24 points - transitive group 24T377
Generators in S24
(3 4)(5 6)(10 12)(13 15)(18 20)(22 24)

(1 8)(2 7)(3 5)(4 6)(9 21)(10 22)(11 23)(12 24)(13 20)(14 19)(15 18)(16 17)

(1 2)(3 4)(5 6)(7 8)(13 15)(14 16)(17 19)(18 20)

(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(21 23)(22 24)

(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 4 2 3)(5 7 6 8)(9 11)(10 12)(13 14 15 16)(17 20 19 18)

(1 9 16)(2 11 14)(3 10 15)(4 12 13)(5 22 18)(6 24 20)(7 23 19)(8 21 17)
 
G:=sub<Sym(24)| (3,4)(5,6)(10,12)(13,15)(18,20)(22,24), (1,8)(2,7)(3,5)(4,6)(9,21)(10,22)(11,23)(12,24)(13,20)(14,19)(15,18)(16,17), (1,2)(3,4)(5,6)(7,8)(13,15)(14,16)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,23)(22,24), (5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,7,6,8)(9,11)(10,12)(13,14,15,16)(17,20,19,18), (1,9,16)(2,11,14)(3,10,15)(4,12,13)(5,22,18)(6,24,20)(7,23,19)(8,21,17)>;
 
G:=Group( (3,4)(5,6)(10,12)(13,15)(18,20)(22,24), (1,8)(2,7)(3,5)(4,6)(9,21)(10,22)(11,23)(12,24)(13,20)(14,19)(15,18)(16,17), (1,2)(3,4)(5,6)(7,8)(13,15)(14,16)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,23)(22,24), (5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,7,6,8)(9,11)(10,12)(13,14,15,16)(17,20,19,18), (1,9,16)(2,11,14)(3,10,15)(4,12,13)(5,22,18)(6,24,20)(7,23,19)(8,21,17) );
 
G=PermutationGroup([[(3,4),(5,6),(10,12),(13,15),(18,20),(22,24)], [(1,8),(2,7),(3,5),(4,6),(9,21),(10,22),(11,23),(12,24),(13,20),(14,19),(15,18),(16,17)], [(1,2),(3,4),(5,6),(7,8),(13,15),(14,16),(17,19),(18,20)], [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12),(21,23),(22,24)], [(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4,2,3),(5,7,6,8),(9,11),(10,12),(13,14,15,16),(17,20,19,18)], [(1,9,16),(2,11,14),(3,10,15),(4,12,13),(5,22,18),(6,24,20),(7,23,19),(8,21,17)]])
 
G:=TransitiveGroup(24,377);
 

►On 24 points - transitive group 24T380
Generators in S24
(1 3)(2 4)(5 8)(6 7)(9 10)(11 12)(13 14)(15 16)(17 18)(19 20)(21 22)(23 24)

(1 8)(2 7)(3 5)(4 6)(9 21)(10 22)(11 23)(12 24)(13 20)(14 19)(15 18)(16 17)

(1 2)(3 4)(5 6)(7 8)(13 15)(14 16)(17 19)(18 20)

(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(21 23)(22 24)

(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 4 2 3)(5 7 6 8)(9 11)(10 12)(13 14 15 16)(17 20 19 18)

(1 9 16)(2 11 14)(3 10 15)(4 12 13)(5 22 18)(6 24 20)(7 23 19)(8 21 17)
 
G:=sub<Sym(24)| (1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24), (1,8)(2,7)(3,5)(4,6)(9,21)(10,22)(11,23)(12,24)(13,20)(14,19)(15,18)(16,17), (1,2)(3,4)(5,6)(7,8)(13,15)(14,16)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,23)(22,24), (5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,7,6,8)(9,11)(10,12)(13,14,15,16)(17,20,19,18), (1,9,16)(2,11,14)(3,10,15)(4,12,13)(5,22,18)(6,24,20)(7,23,19)(8,21,17)>;
 
G:=Group( (1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24), (1,8)(2,7)(3,5)(4,6)(9,21)(10,22)(11,23)(12,24)(13,20)(14,19)(15,18)(16,17), (1,2)(3,4)(5,6)(7,8)(13,15)(14,16)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,23)(22,24), (5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,7,6,8)(9,11)(10,12)(13,14,15,16)(17,20,19,18), (1,9,16)(2,11,14)(3,10,15)(4,12,13)(5,22,18)(6,24,20)(7,23,19)(8,21,17) );
 
G=PermutationGroup([[(1,3),(2,4),(5,8),(6,7),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,22),(23,24)], [(1,8),(2,7),(3,5),(4,6),(9,21),(10,22),(11,23),(12,24),(13,20),(14,19),(15,18),(16,17)], [(1,2),(3,4),(5,6),(7,8),(13,15),(14,16),(17,19),(18,20)], [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12),(21,23),(22,24)], [(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4,2,3),(5,7,6,8),(9,11),(10,12),(13,14,15,16),(17,20,19,18)], [(1,9,16),(2,11,14),(3,10,15),(4,12,13),(5,22,18),(6,24,20),(7,23,19),(8,21,17)]])
 
G:=TransitiveGroup(24,380);
 

►On 24 points - transitive group 24T382
Generators in S24
(1 2)(3 8)(4 7)(5 6)(9 17)(10 20)(11 19)(12 18)(13 24)(14 23)(15 22)(16 21)

(1 8)(2 3)(4 6)(5 7)(9 18)(10 19)(11 20)(12 17)(13 14)(15 16)(21 22)(23 24)

(1 6)(2 5)(3 7)(4 8)(13 15)(14 16)(21 23)(22 24)

(1 6)(2 5)(3 7)(4 8)(9 11)(10 12)(17 19)(18 20)

(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 8 6 4)(2 7 5 3)(9 19)(10 20)(11 17)(12 18)(13 23 15 21)(14 24 16 22)

(1 17 21)(2 9 16)(3 18 15)(4 10 24)(5 11 14)(6 19 23)(7 20 13)(8 12 22)
 
G:=sub<Sym(24)| (1,2)(3,8)(4,7)(5,6)(9,17)(10,20)(11,19)(12,18)(13,24)(14,23)(15,22)(16,21), (1,8)(2,3)(4,6)(5,7)(9,18)(10,19)(11,20)(12,17)(13,14)(15,16)(21,22)(23,24), (1,6)(2,5)(3,7)(4,8)(13,15)(14,16)(21,23)(22,24), (1,6)(2,5)(3,7)(4,8)(9,11)(10,12)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8,6,4)(2,7,5,3)(9,19)(10,20)(11,17)(12,18)(13,23,15,21)(14,24,16,22), (1,17,21)(2,9,16)(3,18,15)(4,10,24)(5,11,14)(6,19,23)(7,20,13)(8,12,22)>;
 
G:=Group( (1,2)(3,8)(4,7)(5,6)(9,17)(10,20)(11,19)(12,18)(13,24)(14,23)(15,22)(16,21), (1,8)(2,3)(4,6)(5,7)(9,18)(10,19)(11,20)(12,17)(13,14)(15,16)(21,22)(23,24), (1,6)(2,5)(3,7)(4,8)(13,15)(14,16)(21,23)(22,24), (1,6)(2,5)(3,7)(4,8)(9,11)(10,12)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8,6,4)(2,7,5,3)(9,19)(10,20)(11,17)(12,18)(13,23,15,21)(14,24,16,22), (1,17,21)(2,9,16)(3,18,15)(4,10,24)(5,11,14)(6,19,23)(7,20,13)(8,12,22) );
 
G=PermutationGroup([[(1,2),(3,8),(4,7),(5,6),(9,17),(10,20),(11,19),(12,18),(13,24),(14,23),(15,22),(16,21)], [(1,8),(2,3),(4,6),(5,7),(9,18),(10,19),(11,20),(12,17),(13,14),(15,16),(21,22),(23,24)], [(1,6),(2,5),(3,7),(4,8),(13,15),(14,16),(21,23),(22,24)], [(1,6),(2,5),(3,7),(4,8),(9,11),(10,12),(17,19),(18,20)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8,6,4),(2,7,5,3),(9,19),(10,20),(11,17),(12,18),(13,23,15,21),(14,24,16,22)], [(1,17,21),(2,9,16),(3,18,15),(4,10,24),(5,11,14),(6,19,23),(7,20,13),(8,12,22)]])
 
G:=TransitiveGroup(24,382);
 

Matrix representation of C24.6A4 ►in GL12(ℤ)

100000000000
010000000000
001000000000
000001000000
000-1-1-1000000
000100000000
000000-1-1-1000
000000001000
000000010000
000000000010
000000000100
000000000-1-1-1
,
100000000000
010000000000
001000000000
000010000000
000100000000
000-1-1-1000000
000000001000
000000-1-1-1000
000000100000
000000000-1-1-1
000000000001
000000000010
,
001000000000
-1-1-1000000000
100000000000
000001000000
000-1-1-1000000
000100000000
000000001000
000000-1-1-1000
000000100000
000000000001
000000000-1-1-1
000000000100
,
010000000000
100000000000
-1-1-1000000000
000010000000
000100000000
000-1-1-1000000
000000010000
000000100000
000000-1-1-1000
000000000010
000000000100
000000000-1-1-1
,
000000010000
000000100000
000000-1-1-1000
000000000010
000000000100
000000000-1-1-1
001000000000
-1-1-1000000000
100000000000
000001000000
000-1-1-1000000
000100000000
,
000100000000
000010000000
000001000000
001000000000
-1-1-1000000000
100000000000
000000000100
000000000010
000000000001
000000001000
000000-1-1-1000
000000100000
,
100000000000
001000000000
-1-1-1000000000
000000100000
000000001000
000000-1-1-1000
000000000010
000000000-1-1-1
000000000001
000001000000
000100000000
000010000000

G:=sub<GL(12,Integers())| [1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0],[0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0],[0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0],[1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0] >;
 

C24.6A4 in GAP, Magma, Sage, TeX 

C_2^4._6A_4
 
% in TeX
 
G:=Group("C2^4.6A4");
 
// GroupNames label
 
G:=SmallGroup(192,1008);
 
// by ID
 
G=gap.SmallGroup(192,1008);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,4371,850,185,360,2524,2111,1173,102,1027,1784]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^3=1,e^2=d*c=g*c*g^-1=c*d,f^2=g*d*g^-1=c,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,a*g=g*a,e*b*e^-1=b*c=c*b,f*b*f^-1=b*d=d*b,b*g=g*b,c*e=e*c,c*f=f*c,g*f*g^-1=d*e=e*d,d*f=f*d,e*f=f*e,g*e*g^-1=c*e*f>;
 
// generators/relations
 

Export

Character table of C24.6A4 in TeX

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