C2^4.2A4 - GroupNames

G = C₂⁴.₂A₄ order 192 = 2⁶·3

2^nd non-split extension by C₂⁴ of A₄ acting faithfully

Aliases: C₂⁴.₂A₄, C₂³.₂SL₂(𝔽₃), C₂³.Q₈⋊C₃, C₂³.₁₇(C₂×A₄), C₂.C₄²⋊₁C₆, C₂³.₃A₄⋊₂C₂, C₂.₂(C₄²⋊C₆), C₂².₃(C₂×SL₂(𝔽₃)), SmallGroup(192,197)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₂.C₄² — C₂⁴.₂A₄

Chief series

C₁ — C₂ — C₂³ — C₂.C₄² — C₂³.₃A₄ — C₂⁴.₂A₄

Lower central

C₂.C₄² — C₂⁴.₂A₄

Upper central

C₁ — C₂

Generators and relations for C₂⁴.₂A₄
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=g³=1, e²=gbg^-1=bcd, f²=gcg^-1=b, ab=ba, ac=ca, ad=da, eae^-1=abd, faf^-1=acd, ag=ga, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fef^-1=de=ed, df=fd, dg=gd, geg^-1=bef, gfg^-1=cde >

C₁

C₂

₃C₂

₄C₂

₁₆C₃

C₂²

₃C₂²

₄C₂²

₆C₂²

₆C₄

₆C₂²

₁₂C₄

₁₂C₂²

₁₆C₆

C₂³

₃C₂×C₄

₆C₂³

₆C₂×C₄

₆C₂³

₆C₂×C₄

₄A₄

₁₆C₂×C₆

C₂⁴

₃C₂²×C₄

₆C₄⋊C₄

₆C₂²⋊C₄

₆C₄⋊C₄

₆C₂²⋊C₄

₄C₂×A₄

C₂.C₄²

₃C₂×C₂²⋊C₄

₃C₂×C₄⋊C₄

₄C₂²×A₄

C₂³.Q₈

C₂³.₃A₄

C₂⁴.₂A₄

Character table of C₂⁴.₂A₄

class	1	2A	2B	2C	2D	2E	3A	3B	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F
size	1	1	3	3	4	4	16	16	12	12	12	12	16	16	16	16	16	16
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	1	1	1	1	-1	-1	-1	1	-1	-1	-1	1	linear of order 2
ρ₃	1	1	1	1	1	1	ζ₃	ζ₃²	1	1	1	1	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃	linear of order 3
ρ₄	1	1	1	1	-1	-1	ζ₃²	ζ₃	1	1	-1	-1	ζ₆⁵	ζ₃	ζ₆	ζ₆	ζ₆⁵	ζ₃²	linear of order 6
ρ₅	1	1	1	1	1	1	ζ₃²	ζ₃	1	1	1	1	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃²	linear of order 3
ρ₆	1	1	1	1	-1	-1	ζ₃	ζ₃²	1	1	-1	-1	ζ₆	ζ₃²	ζ₆⁵	ζ₆⁵	ζ₆	ζ₃	linear of order 6
ρ₇	2	-2	2	-2	2	-2	-1	-1	0	0	0	0	-1	1	1	-1	1	1	symplectic lifted from SL₂(𝔽₃), Schur index 2
ρ₈	2	-2	2	-2	-2	2	-1	-1	0	0	0	0	1	1	-1	1	-1	1	symplectic lifted from SL₂(𝔽₃), Schur index 2
ρ₉	2	-2	2	-2	-2	2	ζ₆	ζ₆⁵	0	0	0	0	ζ₃	ζ₃	ζ₆	ζ₃²	ζ₆⁵	ζ₃²	complex lifted from SL₂(𝔽₃)
ρ₁₀	2	-2	2	-2	-2	2	ζ₆⁵	ζ₆	0	0	0	0	ζ₃²	ζ₃²	ζ₆⁵	ζ₃	ζ₆	ζ₃	complex lifted from SL₂(𝔽₃)
ρ₁₁	2	-2	2	-2	2	-2	ζ₆	ζ₆⁵	0	0	0	0	ζ₆⁵	ζ₃	ζ₃²	ζ₆	ζ₃	ζ₃²	complex lifted from SL₂(𝔽₃)
ρ₁₂	2	-2	2	-2	2	-2	ζ₆⁵	ζ₆	0	0	0	0	ζ₆	ζ₃²	ζ₃	ζ₆⁵	ζ₃²	ζ₃	complex lifted from SL₂(𝔽₃)
ρ₁₃	3	3	3	3	-3	-3	0	0	-1	-1	1	1	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₄	3	3	3	3	3	3	0	0	-1	-1	-1	-1	0	0	0	0	0	0	orthogonal lifted from A₄
ρ₁₅	6	-6	-2	2	0	0	0	0	0	0	2	-2	0	0	0	0	0	0	orthogonal faithful
ρ₁₆	6	-6	-2	2	0	0	0	0	0	0	-2	2	0	0	0	0	0	0	orthogonal faithful
ρ₁₇	6	6	-2	-2	0	0	0	0	2i	-2i	0	0	0	0	0	0	0	0	complex lifted from C₄²⋊C₆
ρ₁₈	6	6	-2	-2	0	0	0	0	-2i	2i	0	0	0	0	0	0	0	0	complex lifted from C₄²⋊C₆

Permutation representations of C₂⁴.₂A₄
►On 12 points - transitive group 12T91

Generators in S₁₂

(1 4)(2 3)(5 6)(7 8)(9 10)(11 12)

(5 7)(6 8)

(1 2)(3 4)

(1 2)(3 4)(5 7)(6 8)(9 11)(10 12)

(3 4)(5 6)(7 8)(9 10 11 12)

(1 3)(2 4)(5 6 7 8)(9 11)

(1 10 7)(2 12 5)(3 11 6)(4 9 8)

G:=sub<Sym(12)| (1,4)(2,3)(5,6)(7,8)(9,10)(11,12), (5,7)(6,8), (1,2)(3,4), (1,2)(3,4)(5,7)(6,8)(9,11)(10,12), (3,4)(5,6)(7,8)(9,10,11,12), (1,3)(2,4)(5,6,7,8)(9,11), (1,10,7)(2,12,5)(3,11,6)(4,9,8)>;

G:=Group( (1,4)(2,3)(5,6)(7,8)(9,10)(11,12), (5,7)(6,8), (1,2)(3,4), (1,2)(3,4)(5,7)(6,8)(9,11)(10,12), (3,4)(5,6)(7,8)(9,10,11,12), (1,3)(2,4)(5,6,7,8)(9,11), (1,10,7)(2,12,5)(3,11,6)(4,9,8) );

G=PermutationGroup([[(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)], [(5,7),(6,8)], [(1,2),(3,4)], [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)], [(3,4),(5,6),(7,8),(9,10,11,12)], [(1,3),(2,4),(5,6,7,8),(9,11)], [(1,10,7),(2,12,5),(3,11,6),(4,9,8)]])

G:=TransitiveGroup(12,91);

►On 12 points - transitive group 12T93

Generators in S₁₂

(3 8)(5 6)(10 12)

(1 2)(5 6)

(3 8)(4 7)

(1 2)(3 8)(4 7)(5 6)(9 11)(10 12)

(3 4)(5 6)(7 8)(9 10 11 12)

(1 5 2 6)(4 7)(9 10)(11 12)

(1 7 9)(2 4 11)(3 12 5)(6 8 10)

G:=sub<Sym(12)| (3,8)(5,6)(10,12), (1,2)(5,6), (3,8)(4,7), (1,2)(3,8)(4,7)(5,6)(9,11)(10,12), (3,4)(5,6)(7,8)(9,10,11,12), (1,5,2,6)(4,7)(9,10)(11,12), (1,7,9)(2,4,11)(3,12,5)(6,8,10)>;

G:=Group( (3,8)(5,6)(10,12), (1,2)(5,6), (3,8)(4,7), (1,2)(3,8)(4,7)(5,6)(9,11)(10,12), (3,4)(5,6)(7,8)(9,10,11,12), (1,5,2,6)(4,7)(9,10)(11,12), (1,7,9)(2,4,11)(3,12,5)(6,8,10) );

G=PermutationGroup([[(3,8),(5,6),(10,12)], [(1,2),(5,6)], [(3,8),(4,7)], [(1,2),(3,8),(4,7),(5,6),(9,11),(10,12)], [(3,4),(5,6),(7,8),(9,10,11,12)], [(1,5,2,6),(4,7),(9,10),(11,12)], [(1,7,9),(2,4,11),(3,12,5),(6,8,10)]])

G:=TransitiveGroup(12,93);

►On 24 points - transitive group 24T460

Generators in S₂₄

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

(5 10)(6 9)(11 15)(12 16)

(1 3)(2 4)(7 8)(13 14)

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

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

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

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

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

G:=TransitiveGroup(24,460);

►On 24 points - transitive group 24T461

Generators in S₂₄

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

(1 6)(2 5)(7 9)(8 10)

(3 12)(4 11)(13 16)(14 15)

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

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

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

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

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

G:=TransitiveGroup(24,461);

►On 24 points - transitive group 24T462

Generators in S₂₄

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

(1 14)(2 13)(3 6)(4 5)

(7 16)(8 15)(9 12)(10 11)

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

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

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

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

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

G:=TransitiveGroup(24,462);

►On 24 points - transitive group 24T467

Generators in S₂₄

(1 14)(3 8)(4 7)(10 15)(18 20)(22 24)

(3 8)(4 7)(5 12)(6 11)

(1 14)(2 13)(9 16)(10 15)

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

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

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

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

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

G:=TransitiveGroup(24,467);

►On 24 points - transitive group 24T468

Generators in S₂₄

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

(1 2)(3 4)(11 12)(15 16)

(5 10)(6 9)(7 14)(8 13)

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

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

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

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

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

G:=TransitiveGroup(24,468);

Polynomial with Galois group C₂⁴.₂A₄ over ℚ

action	f(x)	Disc(f)
12T91	x¹²-534x¹⁰+78489x⁸-4839186x⁶+143348046x⁴-2021896020x²+10884540241	2³⁶·3¹⁶·17³⁰·19³⁰·423557⁴
12T93	x¹²-30x¹⁰+327x⁸-1566x⁶+3096x⁴-1938x²+323	2²⁴·3²⁰·7¹²·17⁵·19⁵

Matrix representation of C₂⁴.₂A₄ ►in GL₆(ℤ)

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	-1	0	0	0
0	0	0	-1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

-1	0	0	0	0	0
0	-1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

-1	0	0	0	0	0
0	-1	0	0	0	0
0	0	-1	0	0	0
0	0	0	-1	0	0
0	0	0	0	-1	0
0	0	0	0	0	-1

1	0	0	0	0	0
0	-1	0	0	0	0
0	0	0	-1	0	0
0	0	-1	0	0	0
0	0	0	0	0	-1
0	0	0	0	1	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	1	0	0
0	0	-1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	-1

0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1
1	0	0	0	0	0
0	1	0	0	0	0

G:=sub<GL(6,Integers())| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂⁴.₂A₄ in GAP, Magma, Sage, TeX

C_2^4._2A_4

% in TeX

G:=Group("C2^4.2A4");

// GroupNames label

G:=SmallGroup(192,197);

// by ID

G=gap.SmallGroup(192,197);

# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,632,135,268,4371,934,521,304,2531,1524]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^3=1,e^2=g*b*g^-1=b*c*d,f^2=g*c*g^-1=b,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*d,f*a*f^-1=a*c*d,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=b*e*f,g*f*g^-1=c*d*e>;

// generators/relations

Export

Subgroup lattice of C₂⁴.₂A₄ in TeX
Character table of C₂⁴.₂A₄ in TeX

G = C24.2A4 order 192 = 26·3

2nd non-split extension by C24 of A4 acting faithfully

G = C₂⁴.₂A₄ order 192 = 2⁶·3

2^nd non-split extension by C₂⁴ of A₄ acting faithfully