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## G = C23×A4order 96 = 25·3

### Direct product of C23 and A4

Aliases: C23×A4, C251C3, C243C6, C233(C2×C6), C22⋊(C22×C6), SmallGroup(96,228)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C23×A4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 454 in 178 conjugacy classes, 48 normal (6 characteristic)
C1, C2 [×7], C2 [×8], C3, C22, C22 [×7], C22 [×49], C6 [×7], C23, C23 [×7], C23 [×49], A4, C2×C6 [×7], C24 [×7], C24 [×8], C2×A4 [×7], C22×C6, C25, C22×A4 [×7], C23×A4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, A4, C2×C6 [×7], C2×A4 [×7], C22×C6, C22×A4 [×7], C23×A4

Permutation representations of C23×A4
On 24 points - transitive group 24T135
Generators in S24
(1 19)(2 20)(3 21)(4 10)(5 11)(6 12)(7 16)(8 17)(9 18)(13 22)(14 23)(15 24)
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 10)(2 11)(3 12)(4 19)(5 20)(6 21)(7 13)(8 14)(9 15)(16 22)(17 23)(18 24)
(2 8)(3 9)(5 23)(6 24)(11 14)(12 15)(17 20)(18 21)
(1 7)(3 9)(4 22)(6 24)(10 13)(12 15)(16 19)(18 21)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)

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

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

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

G:=TransitiveGroup(24,135);

On 24 points - transitive group 24T136
Generators in S24
(1 22)(2 23)(3 24)(4 10)(5 11)(6 12)(7 14)(8 15)(9 13)(16 21)(17 19)(18 20)
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)
(1 15)(2 13)(3 14)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 18)(11 16)(12 17)
(1 15)(2 16)(3 12)(4 20)(5 9)(6 24)(7 19)(8 22)(10 18)(11 13)(14 17)(21 23)
(1 10)(2 13)(3 17)(4 22)(5 21)(6 7)(8 20)(9 23)(11 16)(12 14)(15 18)(19 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)

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

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

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

G:=TransitiveGroup(24,136);

C23×A4 is a maximal subgroup of   C25.S3
C23×A4 is a maximal quotient of   2- 1+43C6

32 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 6A ··· 6N order 1 2 ··· 2 2 ··· 2 3 3 6 ··· 6 size 1 1 ··· 1 3 ··· 3 4 4 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 3 3 type + + + + image C1 C2 C3 C6 A4 C2×A4 kernel C23×A4 C22×A4 C25 C24 C23 C22 # reps 1 7 2 14 1 7

Matrix representation of C23×A4 in GL5(𝔽7)

 1 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6
,
 6 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6
,
 1 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6
,
 1 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 1 0 0 0 0 0 6
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 6
,
 1 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0

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

C23×A4 in GAP, Magma, Sage, TeX

C_2^3\times A_4
% in TeX

G:=Group("C2^3xA4");
// GroupNames label

G:=SmallGroup(96,228);
// by ID

G=gap.SmallGroup(96,228);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,2,202,347]);
// Polycyclic

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

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