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## G = C2×C4×A4order 96 = 25·3

### Direct product of C2×C4 and A4

Aliases: C2×C4×A4, C24.C6, C232C12, C4(C4×A4), (C23×C4)⋊C3, C22⋊(C2×C12), (C22×C4)⋊2C6, C2.1(C22×A4), C23.5(C2×C6), C22.7(C2×A4), (C2×A4).6C22, (C22×A4).2C2, SmallGroup(96,196)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×C4×A4
 Chief series C1 — C22 — C23 — C2×A4 — C22×A4 — C2×C4×A4
 Lower central C22 — C2×C4×A4
 Upper central C1 — C2×C4

Generators and relations for C2×C4×A4
G = < a,b,c,d,e | a2=b4=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 158 in 66 conjugacy classes, 24 normal (12 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, C23, C23, C23, C12, A4, C2×C6, C22×C4, C22×C4, C24, C2×C12, C2×A4, C2×A4, C23×C4, C4×A4, C22×A4, C2×C4×A4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, A4, C2×C6, C2×C12, C2×A4, C4×A4, C22×A4, C2×C4×A4

Permutation representations of C2×C4×A4
On 24 points - transitive group 24T133
Generators in S24
(1 9)(2 10)(3 11)(4 12)(5 24)(6 21)(7 22)(8 23)(13 18)(14 19)(15 20)(16 17)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(5 22)(6 23)(7 24)(8 21)(13 20)(14 17)(15 18)(16 19)
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)
(1 17 8)(2 18 5)(3 19 6)(4 20 7)(9 16 23)(10 13 24)(11 14 21)(12 15 22)

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

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

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

G:=TransitiveGroup(24,133);

On 24 points - transitive group 24T134
Generators in S24
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)(17 23)(18 24)(19 21)(20 22)
(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)(2 4)(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)(17 23)(18 24)(19 21)(20 22)
(1 11)(2 12)(3 9)(4 10)(5 7)(6 8)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 16 23)(10 13 24)(11 14 21)(12 15 22)

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

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

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

G:=TransitiveGroup(24,134);

On 24 points - transitive group 24T147
Generators in S24
(1 23)(2 24)(3 21)(4 22)(5 13)(6 14)(7 15)(8 16)(9 19)(10 20)(11 17)(12 18)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(1 14 11)(2 15 12)(3 16 9)(4 13 10)(5 20 22)(6 17 23)(7 18 24)(8 19 21)

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

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

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

G:=TransitiveGroup(24,147);

C2×C4×A4 is a maximal subgroup of   A4⋊M4(2)  C24.3D6  C24.4D6  C24.5D6  C24.10D6
C2×C4×A4 is a maximal quotient of   (C2×Q8)⋊C12  C4○D4⋊C12  M4(2).A4

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 12A ··· 12H order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 3 3 3 3 4 4 1 1 1 1 3 3 3 3 4 ··· 4 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + + + + + image C1 C2 C2 C3 C4 C6 C6 C12 A4 C2×A4 C2×A4 C4×A4 kernel C2×C4×A4 C4×A4 C22×A4 C23×C4 C2×A4 C22×C4 C24 C23 C2×C4 C4 C22 C2 # reps 1 2 1 2 4 4 2 8 1 2 1 4

Matrix representation of C2×C4×A4 in GL4(𝔽13) generated by

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

C2×C4×A4 in GAP, Magma, Sage, TeX

C_2\times C_4\times A_4
% in TeX

G:=Group("C2xC4xA4");
// GroupNames label

G:=SmallGroup(96,196);
// by ID

G=gap.SmallGroup(96,196);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,2,79,376,665]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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