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G = A4xC4oD4order 192 = 26·3

Direct product of A4 and C4oD4

direct product, metabelian, soluble, monomial

Aliases: A4xC4oD4, (D4xA4):5C2, D4:2(C2xA4), (Q8xA4):5C2, Q8:4(C2xA4), C24.(C2xC6), (C23xC4):3C6, (C22xD4):3C6, C2.6(C23xA4), C4.8(C22xA4), (C22xQ8):7C6, (C2xA4).14C23, (C4xA4).21C22, C22.7(C22xA4), (C22xA4).2C22, C23.31(C22xC6), (C2xC4xA4):7C2, (C2xC4):3(C2xA4), C22:(C3xC4oD4), (C22xC4oD4):2C3, (C22xC4).5(C2xC6), SmallGroup(192,1501)

Series: Derived Chief Lower central Upper central

C1C23 — A4xC4oD4
C1C22C23C2xA4C22xA4C2xC4xA4 — A4xC4oD4
C22C23 — A4xC4oD4
C1C4C4oD4

Generators and relations for A4xC4oD4
 G = < a,b,c,d,e,f | a2=b2=c3=d4=f2=1, e2=d2, cac-1=ab=ba, ad=da, ae=ea, af=fa, cbc-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=d2e >

Subgroups: 620 in 215 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C12, A4, C2xC6, C22xC4, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C4oD4, C24, C2xC12, C3xD4, C3xQ8, C2xA4, C2xA4, C23xC4, C22xD4, C22xQ8, C2xC4oD4, C4xA4, C4xA4, C3xC4oD4, C22xA4, C22xC4oD4, C2xC4xA4, D4xA4, Q8xA4, A4xC4oD4
Quotients: C1, C2, C3, C22, C6, C23, A4, C2xC6, C4oD4, C2xA4, C22xC6, C3xC4oD4, C22xA4, C23xA4, A4xC4oD4

Permutation representations of A4xC4oD4
On 24 points - transitive group 24T296
Generators in S24
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(1 13 11)(2 14 12)(3 15 9)(4 16 10)(5 18 21)(6 19 22)(7 20 23)(8 17 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)
(1 23 3 21)(2 24 4 22)(5 13 7 15)(6 14 8 16)(9 18 11 20)(10 19 12 17)
(5 7)(6 8)(17 19)(18 20)(21 23)(22 24)

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

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

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

G:=TransitiveGroup(24,296);

40 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I3A3B4A4B4C4D4E4F4G4H4I4J6A6B6C···6H12A12B12C12D12E···12J
order1222222222334444444444666···61212121212···12
size1122233666441122233666448···844448···8

40 irreducible representations

dim111111112233336
type++++++++
imageC1C2C2C2C3C6C6C6C4oD4C3xC4oD4A4C2xA4C2xA4C2xA4A4xC4oD4
kernelA4xC4oD4C2xC4xA4D4xA4Q8xA4C22xC4oD4C23xC4C22xD4C22xQ8A4C22C4oD4C2xC4D4Q8C1
# reps133126622413312

Matrix representation of A4xC4oD4 in GL5(F13)

10000
01000
00010
00100
00121212
,
10000
01000
00001
00121212
00100
,
10000
01000
00900
00444
00090
,
80000
08000
001200
000120
000012
,
59000
08000
001200
000120
000012
,
84000
75000
00100
00010
00001

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,1,0,12,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,12,0,0,0,1,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,9,4,0,0,0,0,4,9,0,0,0,4,0],[8,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[5,0,0,0,0,9,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,7,0,0,0,4,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4xC4oD4 in GAP, Magma, Sage, TeX

A_4\times C_4\circ D_4
% in TeX

G:=Group("A4xC4oD4");
// GroupNames label

G:=SmallGroup(192,1501);
// by ID

G=gap.SmallGroup(192,1501);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,2,176,590,530,909]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^4=f^2=1,e^2=d^2,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=d^2*e>;
// generators/relations

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