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G = C23xA4order 96 = 25·3

Direct product of C23 and A4

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

Aliases: C23xA4, C25:1C3, C24:3C6, C23:3(C2xC6), C22:(C22xC6), SmallGroup(96,228)

Series: Derived Chief Lower central Upper central

C1C22 — C23xA4
C1C22A4C2xA4C22xA4 — C23xA4
C22 — C23xA4
C1C23

Generators and relations for C23xA4
 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, C2xC6, C24, C24, C2xA4, C22xC6, C25, C22xA4, C23xA4
Quotients: C1, C2, C3, C22, C6, C23, A4, C2xC6, C2xA4, C22xC6, C22xA4, C23xA4

Permutation representations of C23xA4
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);

C23xA4 is a maximal subgroup of   C25.S3
C23xA4 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++++
imageC1C2C3C6A4C2xA4
kernelC23xA4C22xA4C25C24C23C22
# reps1721417

Matrix representation of C23xA4 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] >;

C23xA4 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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