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G = C23.19(C2×A4)  order 192 = 26·3

12nd non-split extension by C23 of C2×A4 acting via C2×A4/C23=C3

non-abelian, soluble

Aliases: (C22×C4).5A4, C23.19(C2×A4), C22.4(C4.A4), C23.84C23⋊C3, C2.3(C42⋊C6), C23.3A4.1C2, C2.C42.1C6, SmallGroup(192,199)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C2.C42 — C23.19(C2×A4)
Chief series
C1 — C2 — C23 — C2.C42 — C23.3A4 — C23.19(C2×A4)
Lower central
C2.C42 — C23.19(C2×A4)
Upper central
C1 — C2

Generators and relations for C23.19(C2×A4)
 G = < a,b,c,d,e,f,g | a2=b2=c2=g3=1, d2=c, e2=a, f2=gbg-1=abc, ab=ba, ac=ca, ad=da, ae=ea, af=fa, gag-1=b, bc=cb, bd=db, be=eb, bf=fb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, ede-1=bcd, fdf-1=acd, dg=gd, geg-1=abcef, gfg-1=bce >

C1
C2
3C2
3C2
16C3
C22
3C22
3C22
4C4
12C4
12C4
16C6
C23
6C2×C4
6C2×C4
6C2×C4
6C2×C4
6C2×C4
6C2×C4
6C2×C4
4A4
16C12
C22×C4
3C22×C4
3C22×C4
4C2×A4
C2.C42
3C2.C42
3C2.C42
4C4×A4
C23.84C23
C23.3A4
C23.19(C2×A4)

Character table of C23.19(C2×A4)

 class 12A2B2C3A3B4A4B4C4D4E4F6A6B12A12B12C12D
 size 113316164412121212161616161616
ρ1111111111111111111    trivial
ρ2111111-1-1-111-111-1-1-1-1    linear of order 2
ρ31111ζ32ζ3-1-1-111-1ζ3ζ32ζ6ζ6ζ65ζ65    linear of order 6
ρ41111ζ3ζ32-1-1-111-1ζ32ζ3ζ65ζ65ζ6ζ6    linear of order 6
ρ51111ζ32ζ3111111ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ61111ζ3ζ32111111ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ72-2-22-1-12i-2i000011i-ii-i    complex lifted from C4.A4
ρ82-2-22-1-1-2i2i000011-ii-ii    complex lifted from C4.A4
ρ92-2-22ζ65ζ6-2i2i0000ζ32ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    complex lifted from C4.A4
ρ102-2-22ζ6ζ652i-2i0000ζ3ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    complex lifted from C4.A4
ρ112-2-22ζ6ζ65-2i2i0000ζ3ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    complex lifted from C4.A4
ρ122-2-22ζ65ζ62i-2i0000ζ32ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    complex lifted from C4.A4
ρ13333300-3-31-1-11000000    orthogonal lifted from C2×A4
ρ1433330033-1-1-1-1000000    orthogonal lifted from A4
ρ156-62-200002i00-2i000000    complex faithful
ρ166-62-20000-2i002i000000    complex faithful
ρ1766-2-2000002i-2i0000000    complex lifted from C42⋊C6
ρ1866-2-200000-2i2i0000000    complex lifted from C42⋊C6

Permutation representations of C23.19(C2×A4)
►On 24 points - transitive group 24T298
Generators in S24
(13 15)(14 16)(17 19)(18 20)

(1 3)(2 4)(9 11)(10 12)

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

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

(1 16 24)(2 13 21)(3 14 22)(4 15 23)(5 12 20)(6 9 17)(7 10 18)(8 11 19)
 
G:=sub<Sym(24)| (13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(9,11)(10,12), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(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), (1,9)(2,10)(3,11)(4,12)(5,6)(7,8)(13,17,15,19)(14,20,16,18)(21,24)(22,23), (1,4)(2,3)(5,22,7,24)(6,21,8,23)(9,10)(11,12)(13,18)(14,19)(15,20)(16,17), (1,16,24)(2,13,21)(3,14,22)(4,15,23)(5,12,20)(6,9,17)(7,10,18)(8,11,19)>;
 
G:=Group( (13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(9,11)(10,12), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(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), (1,9)(2,10)(3,11)(4,12)(5,6)(7,8)(13,17,15,19)(14,20,16,18)(21,24)(22,23), (1,4)(2,3)(5,22,7,24)(6,21,8,23)(9,10)(11,12)(13,18)(14,19)(15,20)(16,17), (1,16,24)(2,13,21)(3,14,22)(4,15,23)(5,12,20)(6,9,17)(7,10,18)(8,11,19) );
 
G=PermutationGroup([[(13,15),(14,16),(17,19),(18,20)], [(1,3),(2,4),(9,11),(10,12)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(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)], [(1,9),(2,10),(3,11),(4,12),(5,6),(7,8),(13,17,15,19),(14,20,16,18),(21,24),(22,23)], [(1,4),(2,3),(5,22,7,24),(6,21,8,23),(9,10),(11,12),(13,18),(14,19),(15,20),(16,17)], [(1,16,24),(2,13,21),(3,14,22),(4,15,23),(5,12,20),(6,9,17),(7,10,18),(8,11,19)]])
 
G:=TransitiveGroup(24,298);
 

►On 24 points - transitive group 24T310
Generators in S24
(17 19)(18 20)(21 23)(22 24)

(1 3)(2 4)(9 11)(10 12)

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

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

(1 19 6)(2 20 7)(3 17 8)(4 18 5)(9 21 15)(10 22 16)(11 23 13)(12 24 14)
 
G:=sub<Sym(24)| (17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(9,11)(10,12), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(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), (1,9)(2,10)(3,11)(4,12)(5,7)(13,15)(17,23,19,21)(18,22,20,24), (1,3)(5,14,7,16)(6,13,8,15)(10,12)(17,23)(18,24)(19,21)(20,22), (1,19,6)(2,20,7)(3,17,8)(4,18,5)(9,21,15)(10,22,16)(11,23,13)(12,24,14)>;
 
G:=Group( (17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(9,11)(10,12), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(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), (1,9)(2,10)(3,11)(4,12)(5,7)(13,15)(17,23,19,21)(18,22,20,24), (1,3)(5,14,7,16)(6,13,8,15)(10,12)(17,23)(18,24)(19,21)(20,22), (1,19,6)(2,20,7)(3,17,8)(4,18,5)(9,21,15)(10,22,16)(11,23,13)(12,24,14) );
 
G=PermutationGroup([[(17,19),(18,20),(21,23),(22,24)], [(1,3),(2,4),(9,11),(10,12)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(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)], [(1,9),(2,10),(3,11),(4,12),(5,7),(13,15),(17,23,19,21),(18,22,20,24)], [(1,3),(5,14,7,16),(6,13,8,15),(10,12),(17,23),(18,24),(19,21),(20,22)], [(1,19,6),(2,20,7),(3,17,8),(4,18,5),(9,21,15),(10,22,16),(11,23,13),(12,24,14)]])
 
G:=TransitiveGroup(24,310);
 

►On 24 points - transitive group 24T312
Generators in S24
(1 3)(2 4)(9 11)(10 12)

(13 15)(14 16)(17 19)(18 20)

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

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

(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 21 15)(10 22 16)(11 23 13)(12 24 14)
 
G:=sub<Sym(24)| (1,3)(2,4)(9,11)(10,12), (13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(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), (1,9,3,11)(2,12,4,10)(5,24)(6,23)(7,22)(8,21)(13,15)(14,16), (5,24,7,22)(6,23,8,21)(9,11)(10,12)(13,17)(14,20)(15,19)(16,18), (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,21,15)(10,22,16)(11,23,13)(12,24,14)>;
 
G:=Group( (1,3)(2,4)(9,11)(10,12), (13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(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), (1,9,3,11)(2,12,4,10)(5,24)(6,23)(7,22)(8,21)(13,15)(14,16), (5,24,7,22)(6,23,8,21)(9,11)(10,12)(13,17)(14,20)(15,19)(16,18), (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,21,15)(10,22,16)(11,23,13)(12,24,14) );
 
G=PermutationGroup([[(1,3),(2,4),(9,11),(10,12)], [(13,15),(14,16),(17,19),(18,20)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(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)], [(1,9,3,11),(2,12,4,10),(5,24),(6,23),(7,22),(8,21),(13,15),(14,16)], [(5,24,7,22),(6,23,8,21),(9,11),(10,12),(13,17),(14,20),(15,19),(16,18)], [(1,6,19),(2,7,20),(3,8,17),(4,5,18),(9,21,15),(10,22,16),(11,23,13),(12,24,14)]])
 
G:=TransitiveGroup(24,312);
 

Matrix representation of C23.19(C2×A4) ►in GL8(𝔽13)

120000000
012000000
001200000
000120000
00001000
00000100
000000120
000000012
,
120000000
012000000
001200000
000120000
000012000
000001200
00000010
00000001
,
120000000
012000000
00100000
00010000
00001000
00000100
00000010
00000001
,
50000000
05000000
00080000
00500000
00000800
00005000
00000008
00000050
,
62000000
17000000
00080000
00800000
000001200
000012000
00000080
00000008
,
16000000
412000000
00010000
00100000
00008000
00000800
00000005
00000050
,
56000000
312000000
000012000
000001200
000000120
000000012
00100000
00010000

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0],[6,1,0,0,0,0,0,0,2,7,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8],[1,4,0,0,0,0,0,0,6,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0],[5,3,0,0,0,0,0,0,6,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0] >;
 

C23.19(C2×A4) in GAP, Magma, Sage, TeX 

C_2^3._{19}(C_2\times A_4)
 
% in TeX
 
G:=Group("C2^3.19(C2xA4)");
 
// GroupNames label
 
G:=SmallGroup(192,199);
 
// by ID
 
G=gap.SmallGroup(192,199);
 
# by ID
 
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,632,135,268,4371,934,521,304,2531,1524]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^3=1,d^2=c,e^2=a,f^2=g*b*g^-1=a*b*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,g*a*g^-1=b,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,e*d*e^-1=b*c*d,f*d*f^-1=a*c*d,d*g=g*d,g*e*g^-1=a*b*c*e*f,g*f*g^-1=b*c*e>;
 
// generators/relations
 

Export

Subgroup lattice of C23.19(C2×A4) in TeX
Character table of C23.19(C2×A4) in TeX

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