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G = A4×M4(2)  order 192 = 26·3

Direct product of A4 and M4(2)

direct product, metabelian, soluble, monomial

Aliases: A4×M4(2), C24.C12, C8⋊3(C2×A4), (C8×A4)⋊7C2, C4.2(C4×A4), (C4×A4).4C4, (C22×C8)⋊3C6, (C22×C4).C12, C22.6(C4×A4), (C23×C4).2C6, C22⋊(C3×M4(2)), (C22×M4(2))⋊C3, (C22×A4).1C4, C4.13(C22×A4), (C4×A4).23C22, C23.18(C2×C12), C2.9(C2×C4×A4), (C2×C4×A4).8C2, (C2×C4).9(C2×A4), (C2×A4).14(C2×C4), (C22×C4).86(C2×C6), SmallGroup(192,1011)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — A4×M4(2)
Chief series
C1 — C22 — C23 — C22×C4 — C4×A4 — C2×C4×A4 — A4×M4(2)
Lower central
C22 — C23 — A4×M4(2)
Upper central
C1 — C4 — M4(2)

Generators and relations for A4×M4(2)
 G = < a,b,c,d,e | a2=b2=c3=d8=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 224 in 83 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C8, C8, C2×C4, C2×C4, C23, C23, C12, A4, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C24, C2×C12, C2×A4, C2×A4, C22×C8, C2×M4(2), C23×C4, C3×M4(2), C4×A4, C22×A4, C22×M4(2), C8×A4, C2×C4×A4, A4×M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, A4, C2×C6, M4(2), C2×C12, C2×A4, C3×M4(2), C4×A4, C22×A4, C2×C4×A4, A4×M4(2)

Permutation representations of A4×M4(2)
►On 24 points - transitive group 24T297
Generators in S24
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)

(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)

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

(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)
 
G:=sub<Sym(24)| (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (1,10,23)(2,11,24)(3,12,17)(4,13,18)(5,14,19)(6,15,20)(7,16,21)(8,9,22), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)>;
 
G:=Group( (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (1,10,23)(2,11,24)(3,12,17)(4,13,18)(5,14,19)(6,15,20)(7,16,21)(8,9,22), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24) );
 
G=PermutationGroup([[(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24)], [(1,5),(2,6),(3,7),(4,8),(17,21),(18,22),(19,23),(20,24)], [(1,10,23),(2,11,24),(3,12,17),(4,13,18),(5,14,19),(6,15,20),(7,16,21),(8,9,22)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24)]])
 
G:=TransitiveGroup(24,297);
 

40 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F6A6B6C6D8A8B8C8D8E8F8G8H12A12B12C12D12E12F24A···24H
order1222223344444466668888888812121212121224···24
size112336441123364488222266664444888···8

40 irreducible representations

dim111111111122333336
type++++++
imageC1C2C2C3C4C4C6C6C12C12M4(2)C3×M4(2)A4C2×A4C2×A4C4×A4C4×A4A4×M4(2)
kernelA4×M4(2)C8×A4C2×C4×A4C22×M4(2)C4×A4C22×A4C22×C8C23×C4C22×C4C24A4C22M4(2)C8C2×C4C4C22C1
# reps121222424424121222

Matrix representation of A4×M4(2) ►in GL5(𝔽73)

10000
01000
00727272
00001
00010
,
10000
01000
00001
00727272
00100
,
10000
01000
00001
00100
00010
,
046000
10000
004600
000460
000046
,
720000
01000
00100
00010
00001

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,72,0,1,0,0,72,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,72,0,0,0,1,72,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[0,1,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,0,0,46],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;
 

A4×M4(2) in GAP, Magma, Sage, TeX 

A_4\times M_4(2)
 
% in TeX
 
G:=Group("A4xM4(2)");
 
// GroupNames label
 
G:=SmallGroup(192,1011);
 
// by ID
 
G=gap.SmallGroup(192,1011);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,2,365,92,80,1027,1784]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^8=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
 
// generators/relations
 

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