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## G = C12×C3⋊C8order 288 = 25·32

### Direct product of C12 and C3⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C12×C3⋊C8
G = < a,b,c | a12=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 146 in 103 conjugacy classes, 74 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C2×C4, C32, C12, C12, C2×C6, C2×C6, C42, C2×C8, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C2×C12, C4×C8, C3×C12, C62, C2×C3⋊C8, C4×C12, C4×C12, C2×C24, C3×C3⋊C8, C6×C12, C6×C12, C4×C3⋊C8, C4×C24, C6×C3⋊C8, C122, C12×C3⋊C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, Dic3, C12, D6, C2×C6, C42, C2×C8, C3×S3, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C4×C8, C3×Dic3, S3×C6, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, C3×C3⋊C8, S3×C12, C6×Dic3, C4×C3⋊C8, C4×C24, C6×C3⋊C8, Dic3×C12, C12×C3⋊C8

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

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A ··· 4L 6A ··· 6F 6G ··· 6O 8A ··· 8P 12A ··· 12X 12Y ··· 12BH 24A ··· 24AF order 1 2 2 2 3 3 3 3 3 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 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 S3 Dic3 D6 C3×S3 C3⋊C8 C4×S3 C3×Dic3 S3×C6 C3×C3⋊C8 S3×C12 kernel C12×C3⋊C8 C6×C3⋊C8 C122 C4×C3⋊C8 C3×C3⋊C8 C6×C12 C2×C3⋊C8 C4×C12 C3×C12 C3⋊C8 C2×C12 C12 C4×C12 C2×C12 C2×C12 C42 C12 C12 C2×C4 C2×C4 C4 C4 # reps 1 2 1 2 8 4 4 2 16 16 8 32 1 2 1 2 8 4 4 2 16 8

Matrix representation of C12×C3⋊C8 in GL3(𝔽73) generated by

 64 0 0 0 24 0 0 0 24
,
 1 0 0 0 64 0 0 46 8
,
 51 0 0 0 43 27 0 18 30
G:=sub<GL(3,GF(73))| [64,0,0,0,24,0,0,0,24],[1,0,0,0,64,46,0,0,8],[51,0,0,0,43,18,0,27,30] >;

C12×C3⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C12xC3:C8");
// GroupNames label

G:=SmallGroup(288,236);
// by ID

G=gap.SmallGroup(288,236);
# by ID

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

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

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