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## G = C3×A4⋊C8order 288 = 25·32

### Direct product of C3 and A4⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×A4⋊C8
G = < a,b,c,d,e | a3=b2=c2=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >

Subgroups: 166 in 58 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C23, C32, C12, C12, A4, A4, C2×C6, C2×C6, C2×C8, C22×C4, C3×C6, C3⋊C8, C24, C2×C12, C2×A4, C2×A4, C22×C6, C22⋊C8, C3×C12, C3×A4, C2×C24, C4×A4, C4×A4, C22×C12, C3×C3⋊C8, C6×A4, C3×C22⋊C8, A4⋊C8, C12×A4, C3×A4⋊C8
Quotients: C1, C2, C3, C4, S3, C6, C8, Dic3, C12, C3×S3, C3⋊C8, C24, S4, C3×Dic3, A4⋊C4, C3×C3⋊C8, C3×S4, A4⋊C8, C3×A4⋊C4, C3×A4⋊C8

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

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 8A ··· 8H 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12N 24A ··· 24P order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 8 ··· 8 12 12 12 12 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 3 3 1 1 8 8 8 1 1 3 3 1 1 3 3 3 3 8 8 8 6 ··· 6 1 1 1 1 3 3 3 3 8 ··· 8 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 type + + + - + image C1 C2 C3 C4 C6 C8 C12 C24 S3 Dic3 C3×S3 C3⋊C8 C3×Dic3 C3×C3⋊C8 S4 A4⋊C4 C3×S4 A4⋊C8 C3×A4⋊C4 C3×A4⋊C8 kernel C3×A4⋊C8 C12×A4 A4⋊C8 C6×A4 C4×A4 C3×A4 C2×A4 A4 C22×C12 C22×C6 C22×C4 C2×C6 C23 C22 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 1 1 2 2 2 4 2 2 4 4 4 8

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

 64 0 0 0 0 0 64 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 72
,
 72 72 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 51 0 0 0 0 22 22 0 0 0 0 0 0 0 63 0 0 0 63 0 0 0 63 0 0

G:=sub<GL(5,GF(73))| [64,0,0,0,0,0,64,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72],[72,1,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[51,22,0,0,0,0,22,0,0,0,0,0,0,0,63,0,0,0,63,0,0,0,63,0,0] >;

C3×A4⋊C8 in GAP, Magma, Sage, TeX

C_3\times A_4\rtimes C_8
% in TeX

G:=Group("C3xA4:C8");
// GroupNames label

G:=SmallGroup(288,398);
// by ID

G=gap.SmallGroup(288,398);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-3,-2,2,42,58,1684,6053,285,3534,475]);
// Polycyclic

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

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