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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×M4(2))⋊C3, C22⋊(C3×M4(2)), (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)

Subgroups: 224 in 83 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22 [×2], C22 [×7], C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×9], C23, C23 [×4], C12 [×2], A4, C2×C6, C2×C8 [×4], M4(2), M4(2) [×5], C22×C4 [×2], C22×C4 [×4], C24, C24 [×2], C2×C12, C2×A4, C2×A4, C22×C8 [×2], C2×M4(2) [×4], C23×C4, C3×M4(2), C4×A4 [×2], C22×A4, C22×M4(2), C8×A4 [×2], C2×C4×A4, A4×M4(2)

Quotients:
C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, C12 [×2], A4, C2×C6, M4(2), C2×C12, C2×A4 [×3], C3×M4(2), C4×A4 [×2], C22×A4, C2×C4×A4, A4×M4(2)

Generators and relations
 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 >

Permutation representations
►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 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 17)(8 9 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)

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

Matrix representation ►G ⊆ 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] >;
 

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

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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