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G = C23×A4  order 96 = 25·3

Direct product of C23 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: C23×A4, C25⋊1C3, C24⋊3C6, C23⋊3(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, C2, C3, C22, C22, C22, C6, C23, C23, C23, A4, C2×C6, C24, C24, C2×A4, C22×C6, C25, C22×A4, C23×A4
Quotients: C1, C2, C3, C22, C6, C23, A4, C2×C6, C2×A4, C22×C6, C22×A4, 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+4⋊3C6

32 conjugacy classes

class 1 2A···2G2H···2O3A3B6A···6N
order12···22···2336···6
size11···13···3444···4

32 irreducible representations

dim111133
type++++
imageC1C2C3C6A4C2×A4
kernelC23×A4C22×A4C25C24C23C22
# reps1721417

Matrix representation of C23×A4 ►in GL5(𝔽7)

10000
01000
00600
00060
00006
,
60000
06000
00600
00060
00006
,
10000
06000
00600
00060
00006
,
10000
01000
00600
00010
00006
,
10000
01000
00100
00060
00006
,
10000
04000
00010
00001
00100

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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