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## G = A4×C3⋊D4order 288 = 25·32

### Direct product of A4 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — A4×C3⋊D4
 Chief series C1 — C3 — C2×C6 — C22×C6 — C6×A4 — C2×S3×A4 — A4×C3⋊D4
 Lower central C2×C6 — C22×C6 — A4×C3⋊D4
 Upper central C1 — C2 — C22

Generators and relations for A4×C3⋊D4
G = < a,b,c,d,e,f | a2=b2=c3=d3=e4=f2=1, 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, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 698 in 142 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C4 [×2], C22 [×2], C22 [×13], S3 [×2], C6, C6 [×10], C2×C4 [×2], D4 [×6], C23, C23 [×8], C32, Dic3, Dic3, C12, A4, A4, D6, D6 [×5], C2×C6 [×2], C2×C6 [×10], C22×C4, C2×D4 [×4], C24, C24, C3×S3, C3×C6 [×2], C2×Dic3 [×2], C3⋊D4, C3⋊D4 [×5], C3×D4, C2×A4, C2×A4 [×5], C22×S3 [×4], C22×C6, C22×C6 [×4], C22×D4, C3×Dic3, C3×A4, S3×C6, C62, C4×A4, C22×Dic3, C2×C3⋊D4 [×4], C22×A4, C22×A4 [×2], S3×C23, C23×C6, C3×C3⋊D4, S3×A4, C6×A4, C6×A4, D4×A4, C22×C3⋊D4, Dic3×A4, C2×S3×A4, A4×C2×C6, A4×C3⋊D4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, A4, D6, C2×C6, C3×S3, C3⋊D4, C3×D4, C2×A4 [×3], S3×C6, C22×A4, C3×C3⋊D4, S3×A4, D4×A4, C2×S3×A4, A4×C3⋊D4

Smallest permutation representation of A4×C3⋊D4
On 36 points
Generators in S36
(1 3)(2 4)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(5 7)(6 8)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(33 35)(34 36)
(1 26 33)(2 27 34)(3 28 35)(4 25 36)(5 31 21)(6 32 22)(7 29 23)(8 30 24)(9 14 19)(10 15 20)(11 16 17)(12 13 18)
(1 15 31)(2 32 16)(3 13 29)(4 30 14)(5 33 10)(6 11 34)(7 35 12)(8 9 36)(17 27 22)(18 23 28)(19 25 24)(20 21 26)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)
(2 4)(5 10)(6 9)(7 12)(8 11)(13 29)(14 32)(15 31)(16 30)(17 24)(18 23)(19 22)(20 21)(25 27)(34 36)

G:=sub<Sym(36)| (1,3)(2,4)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (5,7)(6,8)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(33,35)(34,36), (1,26,33)(2,27,34)(3,28,35)(4,25,36)(5,31,21)(6,32,22)(7,29,23)(8,30,24)(9,14,19)(10,15,20)(11,16,17)(12,13,18), (1,15,31)(2,32,16)(3,13,29)(4,30,14)(5,33,10)(6,11,34)(7,35,12)(8,9,36)(17,27,22)(18,23,28)(19,25,24)(20,21,26), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (2,4)(5,10)(6,9)(7,12)(8,11)(13,29)(14,32)(15,31)(16,30)(17,24)(18,23)(19,22)(20,21)(25,27)(34,36)>;

G:=Group( (1,3)(2,4)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (5,7)(6,8)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(33,35)(34,36), (1,26,33)(2,27,34)(3,28,35)(4,25,36)(5,31,21)(6,32,22)(7,29,23)(8,30,24)(9,14,19)(10,15,20)(11,16,17)(12,13,18), (1,15,31)(2,32,16)(3,13,29)(4,30,14)(5,33,10)(6,11,34)(7,35,12)(8,9,36)(17,27,22)(18,23,28)(19,25,24)(20,21,26), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (2,4)(5,10)(6,9)(7,12)(8,11)(13,29)(14,32)(15,31)(16,30)(17,24)(18,23)(19,22)(20,21)(25,27)(34,36) );

G=PermutationGroup([(1,3),(2,4),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(5,7),(6,8),(9,11),(10,12),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(33,35),(34,36)], [(1,26,33),(2,27,34),(3,28,35),(4,25,36),(5,31,21),(6,32,22),(7,29,23),(8,30,24),(9,14,19),(10,15,20),(11,16,17),(12,13,18)], [(1,15,31),(2,32,16),(3,13,29),(4,30,14),(5,33,10),(6,11,34),(7,35,12),(8,9,36),(17,27,22),(18,23,28),(19,25,24),(20,21,26)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36)], [(2,4),(5,10),(6,9),(7,12),(8,11),(13,29),(14,32),(15,31),(16,30),(17,24),(18,23),(19,22),(20,21),(25,27),(34,36)])

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J ··· 6Q 6R 6S 12A 12B order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 ··· 6 6 6 12 12 size 1 1 2 3 3 6 6 18 2 4 4 8 8 6 18 2 2 2 4 4 6 6 6 6 8 ··· 8 24 24 24 24

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 6 6 6 6 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×C3⋊D4 A4 C2×A4 C2×A4 C2×A4 S3×A4 D4×A4 C2×S3×A4 A4×C3⋊D4 kernel A4×C3⋊D4 Dic3×A4 C2×S3×A4 A4×C2×C6 C22×C3⋊D4 C22×Dic3 S3×C23 C23×C6 C22×A4 C3×A4 C2×A4 C24 A4 C2×C6 C23 C22 C3⋊D4 Dic3 D6 C2×C6 C22 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 4 1 1 1 1 1 1 1 2

Matrix representation of A4×C3⋊D4 in GL5(𝔽13)

 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 12 0 0 0 0 12 0 1 0 0 12 1 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
,
 9 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 12 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[9,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4×C3⋊D4 in GAP, Magma, Sage, TeX

A_4\times C_3\rtimes D_4
% in TeX

G:=Group("A4xC3:D4");
// GroupNames label

G:=SmallGroup(288,928);
// by ID

G=gap.SmallGroup(288,928);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-3,197,648,271,9414]);
// Polycyclic

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

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