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## G = C3×C12⋊2Q8order 288 = 25·32

### Direct product of C3 and C12⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C12⋊2Q8
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — C6×Dic6 — C3×C12⋊2Q8
 Lower central C3 — C2×C6 — C3×C12⋊2Q8
 Upper central C1 — C2×C6 — C4×C12

Generators and relations for C3×C122Q8
G = < a,b,c,d | a3=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 306 in 151 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C4⋊Q8, C3×Dic3, C3×C12, C62, C4⋊Dic3, C4×C12, C4×C12, C3×C4⋊C4, C2×Dic6, C6×Q8, C3×Dic6, C6×Dic3, C6×C12, C6×C12, C122Q8, C3×C4⋊Q8, C3×C4⋊Dic3, C122, C6×Dic6, C3×C122Q8
Quotients: C1, C2, C3, C22, S3, C6, D4, Q8, C23, D6, C2×C6, C2×D4, C2×Q8, C3×S3, Dic6, D12, C3×D4, C3×Q8, C22×S3, C22×C6, C4⋊Q8, S3×C6, C2×Dic6, C2×D12, C6×D4, C6×Q8, C3×Dic6, C3×D12, S3×C2×C6, C122Q8, C3×C4⋊Q8, C6×Dic6, C6×D12, C3×C122Q8

Smallest permutation representation of C3×C122Q8
On 96 points
Generators in S96
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(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 81 21 87)(2 82 22 88)(3 83 23 89)(4 84 24 90)(5 73 13 91)(6 74 14 92)(7 75 15 93)(8 76 16 94)(9 77 17 95)(10 78 18 96)(11 79 19 85)(12 80 20 86)(25 67 44 56)(26 68 45 57)(27 69 46 58)(28 70 47 59)(29 71 48 60)(30 72 37 49)(31 61 38 50)(32 62 39 51)(33 63 40 52)(34 64 41 53)(35 65 42 54)(36 66 43 55)
(1 36 21 43)(2 35 22 42)(3 34 23 41)(4 33 24 40)(5 32 13 39)(6 31 14 38)(7 30 15 37)(8 29 16 48)(9 28 17 47)(10 27 18 46)(11 26 19 45)(12 25 20 44)(49 93 72 75)(50 92 61 74)(51 91 62 73)(52 90 63 84)(53 89 64 83)(54 88 65 82)(55 87 66 81)(56 86 67 80)(57 85 68 79)(58 96 69 78)(59 95 70 77)(60 94 71 76)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A ··· 4F 4G 4H 4I 4J 6A ··· 6F 6G ··· 6O 12A ··· 12AV 12AW ··· 12BD order 1 2 2 2 3 3 3 3 3 4 ··· 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 2 2 2 2 ··· 2 12 12 12 12 1 ··· 1 2 ··· 2 2 ··· 2 12 ··· 12

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 Q8 D6 C3×S3 Dic6 D12 C3×D4 C3×Q8 S3×C6 C3×Dic6 C3×D12 kernel C3×C12⋊2Q8 C3×C4⋊Dic3 C122 C6×Dic6 C12⋊2Q8 C4⋊Dic3 C4×C12 C2×Dic6 C4×C12 C3×C12 C3×C12 C2×C12 C42 C12 C12 C12 C12 C2×C4 C4 C4 # reps 1 4 1 2 2 8 2 4 1 2 4 3 2 8 4 4 8 6 16 8

Matrix representation of C3×C122Q8 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 11 0 0 0 0 6 0 0 0 0 11 0 0 0 0 6
,
 8 0 0 0 0 5 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 12 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[11,0,0,0,0,6,0,0,0,0,11,0,0,0,0,6],[8,0,0,0,0,5,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3×C122Q8 in GAP, Magma, Sage, TeX

C_3\times C_{12}\rtimes_2Q_8
% in TeX

G:=Group("C3xC12:2Q8");
// GroupNames label

G:=SmallGroup(288,640);
// by ID

G=gap.SmallGroup(288,640);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,701,344,590,142,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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