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## G = A4×C3⋊C8order 288 = 25·32

### Direct product of A4 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — A4×C3⋊C8
 Chief series C1 — C3 — C2×C6 — C22×C6 — C22×C12 — C12×A4 — A4×C3⋊C8
 Lower central C2×C6 — A4×C3⋊C8
 Upper central C1 — C4

Generators and relations for A4×C3⋊C8
G = < a,b,c,d,e | a2=b2=c3=d3=e8=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 178 in 58 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×2], C3, C3 [×2], C4, C4, C22, C22 [×2], C6, C6 [×4], C8 [×2], C2×C4 [×2], C23, C32, C12, C12 [×3], A4, A4, C2×C6, C2×C6 [×2], C2×C8 [×2], C22×C4, C3×C6, C3⋊C8, C3⋊C8, C24, C2×C12 [×2], C2×A4, C2×A4, C22×C6, C22×C8, C3×C12, C3×A4, C2×C3⋊C8 [×2], C4×A4, C4×A4, C22×C12, C3×C3⋊C8, C6×A4, C8×A4, C22×C3⋊C8, C12×A4, A4×C3⋊C8
Quotients: C1, C2, C3, C4, S3, C6, C8, Dic3, C12, A4, C3×S3, C3⋊C8, C24, C2×A4, C3×Dic3, C4×A4, C3×C3⋊C8, S3×A4, C8×A4, Dic3×A4, A4×C3⋊C8

Smallest permutation representation of A4×C3⋊C8
On 72 points
Generators in S72
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(57 61)(58 62)(59 63)(60 64)
(1 15 24)(2 16 17)(3 9 18)(4 10 19)(5 11 20)(6 12 21)(7 13 22)(8 14 23)(25 67 41)(26 68 42)(27 69 43)(28 70 44)(29 71 45)(30 72 46)(31 65 47)(32 66 48)(33 55 63)(34 56 64)(35 49 57)(36 50 58)(37 51 59)(38 52 60)(39 53 61)(40 54 62)
(1 49 71)(2 72 50)(3 51 65)(4 66 52)(5 53 67)(6 68 54)(7 55 69)(8 70 56)(9 59 47)(10 48 60)(11 61 41)(12 42 62)(13 63 43)(14 44 64)(15 57 45)(16 46 58)(17 30 36)(18 37 31)(19 32 38)(20 39 25)(21 26 40)(22 33 27)(23 28 34)(24 35 29)
(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 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)

G:=sub<Sym(72)| (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(57,61)(58,62)(59,63)(60,64), (1,15,24)(2,16,17)(3,9,18)(4,10,19)(5,11,20)(6,12,21)(7,13,22)(8,14,23)(25,67,41)(26,68,42)(27,69,43)(28,70,44)(29,71,45)(30,72,46)(31,65,47)(32,66,48)(33,55,63)(34,56,64)(35,49,57)(36,50,58)(37,51,59)(38,52,60)(39,53,61)(40,54,62), (1,49,71)(2,72,50)(3,51,65)(4,66,52)(5,53,67)(6,68,54)(7,55,69)(8,70,56)(9,59,47)(10,48,60)(11,61,41)(12,42,62)(13,63,43)(14,44,64)(15,57,45)(16,46,58)(17,30,36)(18,37,31)(19,32,38)(20,39,25)(21,26,40)(22,33,27)(23,28,34)(24,35,29), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)>;

G:=Group( (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(57,61)(58,62)(59,63)(60,64), (1,15,24)(2,16,17)(3,9,18)(4,10,19)(5,11,20)(6,12,21)(7,13,22)(8,14,23)(25,67,41)(26,68,42)(27,69,43)(28,70,44)(29,71,45)(30,72,46)(31,65,47)(32,66,48)(33,55,63)(34,56,64)(35,49,57)(36,50,58)(37,51,59)(38,52,60)(39,53,61)(40,54,62), (1,49,71)(2,72,50)(3,51,65)(4,66,52)(5,53,67)(6,68,54)(7,55,69)(8,70,56)(9,59,47)(10,48,60)(11,61,41)(12,42,62)(13,63,43)(14,44,64)(15,57,45)(16,46,58)(17,30,36)(18,37,31)(19,32,38)(20,39,25)(21,26,40)(22,33,27)(23,28,34)(24,35,29), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72) );

G=PermutationGroup([(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72)], [(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(57,61),(58,62),(59,63),(60,64)], [(1,15,24),(2,16,17),(3,9,18),(4,10,19),(5,11,20),(6,12,21),(7,13,22),(8,14,23),(25,67,41),(26,68,42),(27,69,43),(28,70,44),(29,71,45),(30,72,46),(31,65,47),(32,66,48),(33,55,63),(34,56,64),(35,49,57),(36,50,58),(37,51,59),(38,52,60),(39,53,61),(40,54,62)], [(1,49,71),(2,72,50),(3,51,65),(4,66,52),(5,53,67),(6,68,54),(7,55,69),(8,70,56),(9,59,47),(10,48,60),(11,61,41),(12,42,62),(13,63,43),(14,44,64),(15,57,45),(16,46,58),(17,30,36),(18,37,31),(19,32,38),(20,39,25),(21,26,40),(22,33,27),(23,28,34),(24,35,29)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72)])

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 24A ··· 24H order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 3 3 2 4 4 8 8 1 1 3 3 2 4 4 6 6 8 8 3 3 3 3 9 9 9 9 2 2 4 4 4 4 6 6 8 8 8 8 12 ··· 12

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 6 6 6 type + + + - + + + - image C1 C2 C3 C4 C6 C8 C12 C24 S3 Dic3 C3×S3 C3⋊C8 C3×Dic3 C3×C3⋊C8 A4 C2×A4 C4×A4 C8×A4 S3×A4 Dic3×A4 A4×C3⋊C8 kernel A4×C3⋊C8 C12×A4 C22×C3⋊C8 C6×A4 C22×C12 C3×A4 C22×C6 C2×C6 C4×A4 C2×A4 C22×C4 A4 C23 C22 C3⋊C8 C12 C6 C3 C4 C2 C1 # reps 1 1 2 2 2 4 4 8 1 1 2 2 2 4 1 1 2 4 1 1 2

Matrix representation of A4×C3⋊C8 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 72 64 0 0 0 0 1 0 0 0 0 0 72
,
 1 0 0 0 0 0 1 0 0 0 0 0 72 0 65 0 0 0 72 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 64 0 0 0 0 2 9 8 0 0 0 72 0
,
 72 72 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 5 42 0 0 0 37 68 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,64,1,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,65,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,64,2,0,0,0,0,9,72,0,0,0,8,0],[72,1,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,37,0,0,0,42,68,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72] >;

A4×C3⋊C8 in GAP, Magma, Sage, TeX

A_4\times C_3\rtimes C_8
% in TeX

G:=Group("A4xC3:C8");
// GroupNames label

G:=SmallGroup(288,408);
// by ID

G=gap.SmallGroup(288,408);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,2,-3,42,58,1271,516,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=e^8=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^-1=d^-1>;
// generators/relations

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