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G = A4×C4○D4order 192 = 26·3

Direct product of A4 and C4○D4

direct product, metabelian, soluble, monomial

Aliases: A4×C4○D4, (D4×A4)⋊5C2, D42(C2×A4), (Q8×A4)⋊5C2, Q84(C2×A4), C24.(C2×C6), (C23×C4)⋊3C6, (C22×D4)⋊3C6, C2.6(C23×A4), C4.8(C22×A4), (C22×Q8)⋊7C6, (C2×A4).14C23, (C4×A4).21C22, C22.7(C22×A4), (C22×A4).2C22, C23.31(C22×C6), (C2×C4×A4)⋊7C2, (C2×C4)⋊3(C2×A4), C22⋊(C3×C4○D4), (C22×C4○D4)⋊2C3, (C22×C4).5(C2×C6), SmallGroup(192,1501)

Series: Derived Chief Lower central Upper central

C1C23 — A4×C4○D4
C1C22C23C2×A4C22×A4C2×C4×A4 — A4×C4○D4
C22C23 — A4×C4○D4
C1C4C4○D4

Generators and relations for A4×C4○D4
 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 [×8], C3, C4, C4 [×3], C4 [×4], C22, C22 [×3], C22 [×17], C6 [×4], C2×C4 [×3], C2×C4 [×23], D4 [×3], D4 [×15], Q8, Q8 [×5], C23, C23 [×12], C12 [×4], A4, C2×C6 [×3], C22×C4, C22×C4 [×3], C22×C4 [×12], C2×D4 [×12], C2×Q8 [×4], C4○D4, C4○D4 [×21], C24 [×3], C2×C12 [×3], C3×D4 [×3], C3×Q8, C2×A4, C2×A4 [×3], C23×C4 [×3], C22×D4 [×3], C22×Q8, C2×C4○D4 [×8], C4×A4, C4×A4 [×3], C3×C4○D4, C22×A4 [×3], C22×C4○D4, C2×C4×A4 [×3], D4×A4 [×3], Q8×A4, A4×C4○D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, A4, C2×C6 [×7], C4○D4, C2×A4 [×7], C22×C6, C3×C4○D4, C22×A4 [×7], C23×A4, A4×C4○D4

Permutation representations of A4×C4○D4
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 9)(2 14 10)(3 15 11)(4 16 12)(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 20 11 18)(10 17 12 19)
(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), (1,13,9)(2,14,10)(3,15,11)(4,16,12)(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)(9,20,11,18)(10,17,12,19), (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), (1,13,9)(2,14,10)(3,15,11)(4,16,12)(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)(9,20,11,18)(10,17,12,19), (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)], [(1,13,9),(2,14,10),(3,15,11),(4,16,12),(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),(9,20,11,18),(10,17,12,19)], [(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++++++++
imageC1C2C2C2C3C6C6C6C4○D4C3×C4○D4A4C2×A4C2×A4C2×A4A4×C4○D4
kernelA4×C4○D4C2×C4×A4D4×A4Q8×A4C22×C4○D4C23×C4C22×D4C22×Q8A4C22C4○D4C2×C4D4Q8C1
# reps133126622413312

Matrix representation of A4×C4○D4 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] >;

A4×C4○D4 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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