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## G = A4×C4○D4order 192 = 26·3

### Direct product of A4 and C4○D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — A4×C4○D4
 Chief series C1 — C22 — C23 — C2×A4 — C22×A4 — C2×C4×A4 — A4×C4○D4
 Lower central C22 — C23 — A4×C4○D4
 Upper central C1 — C4 — C4○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 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C ··· 6H 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 4 4 6 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 2 2 2 3 3 6 6 6 4 4 1 1 2 2 2 3 3 6 6 6 4 4 8 ··· 8 4 4 4 4 8 ··· 8

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 3 3 3 3 6 type + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C4○D4 C3×C4○D4 A4 C2×A4 C2×A4 C2×A4 A4×C4○D4 kernel A4×C4○D4 C2×C4×A4 D4×A4 Q8×A4 C22×C4○D4 C23×C4 C22×D4 C22×Q8 A4 C22 C4○D4 C2×C4 D4 Q8 C1 # reps 1 3 3 1 2 6 6 2 2 4 1 3 3 1 2

Matrix representation of A4×C4○D4 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 12 12 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 12 12 12 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 4 4 4 0 0 0 9 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 9 0 0 0 0 8 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 8 4 0 0 0 7 5 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

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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