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## G = C3×C12⋊C8order 288 = 25·32

### Direct product of C3 and C12⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C12⋊C8
G = < a,b,c | a3=b12=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 146 in 91 conjugacy classes, 58 normal (46 characteristic)
C1, C2, C3, C3, C4, C4, C4, C22, C6, C6, C8, C2×C4, C32, C12, C12, C12, C2×C6, C2×C6, C42, C2×C8, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C4⋊C8, C3×C12, C3×C12, C3×C12, C62, C2×C3⋊C8, C4×C12, C4×C12, C2×C24, C3×C3⋊C8, C6×C12, C12⋊C8, C3×C4⋊C8, C6×C3⋊C8, C122, C3×C12⋊C8
Quotients:

Smallest permutation representation of C3×C12⋊C8
On 96 points
Generators in S96
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 89 93)(86 90 94)(87 91 95)(88 92 96)
(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)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 70 58 23 40 31 96 80)(2 69 59 22 41 30 85 79)(3 68 60 21 42 29 86 78)(4 67 49 20 43 28 87 77)(5 66 50 19 44 27 88 76)(6 65 51 18 45 26 89 75)(7 64 52 17 46 25 90 74)(8 63 53 16 47 36 91 73)(9 62 54 15 48 35 92 84)(10 61 55 14 37 34 93 83)(11 72 56 13 38 33 94 82)(12 71 57 24 39 32 95 81)

G:=sub<Sym(96)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,70,58,23,40,31,96,80)(2,69,59,22,41,30,85,79)(3,68,60,21,42,29,86,78)(4,67,49,20,43,28,87,77)(5,66,50,19,44,27,88,76)(6,65,51,18,45,26,89,75)(7,64,52,17,46,25,90,74)(8,63,53,16,47,36,91,73)(9,62,54,15,48,35,92,84)(10,61,55,14,37,34,93,83)(11,72,56,13,38,33,94,82)(12,71,57,24,39,32,95,81)>;

G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,70,58,23,40,31,96,80)(2,69,59,22,41,30,85,79)(3,68,60,21,42,29,86,78)(4,67,49,20,43,28,87,77)(5,66,50,19,44,27,88,76)(6,65,51,18,45,26,89,75)(7,64,52,17,46,25,90,74)(8,63,53,16,47,36,91,73)(9,62,54,15,48,35,92,84)(10,61,55,14,37,34,93,83)(11,72,56,13,38,33,94,82)(12,71,57,24,39,32,95,81) );

G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,89,93),(86,90,94),(87,91,95),(88,92,96)], [(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),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,70,58,23,40,31,96,80),(2,69,59,22,41,30,85,79),(3,68,60,21,42,29,86,78),(4,67,49,20,43,28,87,77),(5,66,50,19,44,27,88,76),(6,65,51,18,45,26,89,75),(7,64,52,17,46,25,90,74),(8,63,53,16,47,36,91,73),(9,62,54,15,48,35,92,84),(10,61,55,14,37,34,93,83),(11,72,56,13,38,33,94,82),(12,71,57,24,39,32,95,81)]])

108 conjugacy classes

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

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - - + - + image C1 C2 C2 C3 C4 C6 C6 C8 C12 C24 S3 D4 Q8 Dic3 D6 M4(2) C3×S3 C3⋊C8 Dic6 D12 C3×D4 C3×Q8 C3×Dic3 S3×C6 C4.Dic3 C3×M4(2) C3×C3⋊C8 C3×Dic6 C3×D12 C3×C4.Dic3 kernel C3×C12⋊C8 C6×C3⋊C8 C122 C12⋊C8 C6×C12 C2×C3⋊C8 C4×C12 C3×C12 C2×C12 C12 C4×C12 C3×C12 C3×C12 C2×C12 C2×C12 C3×C6 C42 C12 C12 C12 C12 C12 C2×C4 C2×C4 C6 C6 C4 C4 C4 C2 # reps 1 2 1 2 4 4 2 8 8 16 1 1 1 2 1 2 2 4 2 2 2 2 4 2 4 4 8 4 4 8

Matrix representation of C3×C12⋊C8 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 65 0 0 0 0 9 0 0 0 0 3 56 0 0 0 49
,
 0 1 0 0 46 0 0 0 0 0 46 38 0 0 43 27
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[65,0,0,0,0,9,0,0,0,0,3,0,0,0,56,49],[0,46,0,0,1,0,0,0,0,0,46,43,0,0,38,27] >;

C3×C12⋊C8 in GAP, Magma, Sage, TeX

C_3\times C_{12}\rtimes C_8
% in TeX

G:=Group("C3xC12:C8");
// GroupNames label

G:=SmallGroup(288,238);
// by ID

G=gap.SmallGroup(288,238);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,176,136,9414]);
// Polycyclic

G:=Group<a,b,c|a^3=b^12=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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