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## G = C3×C12.6Q8order 288 = 25·32

### Direct product of C3 and C12.6Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C12.6Q8
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — C3×Dic3⋊C4 — C3×C12.6Q8
 Lower central C3 — C2×C6 — C3×C12.6Q8
 Upper central C1 — C2×C6 — C4×C12

Generators and relations for C3×C12.6Q8
G = < a,b,c,d | a3=b12=c4=1, d2=b6c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b6c-1 >

Subgroups: 242 in 127 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×6], C22, C6 [×2], C6 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], C32, Dic3 [×4], C12 [×4], C12 [×14], C2×C6 [×2], C2×C6, C42, C4⋊C4 [×6], C3×C6, C3×C6 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×7], C42.C2, C3×Dic3 [×4], C3×C12 [×2], C3×C12 [×2], C62, Dic3⋊C4 [×4], C4⋊Dic3 [×2], C4×C12 [×2], C4×C12, C3×C4⋊C4 [×6], C6×Dic3 [×4], C6×C12, C6×C12 [×2], C12.6Q8, C3×C42.C2, C3×Dic3⋊C4 [×4], C3×C4⋊Dic3 [×2], C122, C3×C12.6Q8
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], Q8 [×2], C23, D6 [×3], C2×C6 [×7], C2×Q8, C4○D4 [×2], C3×S3, Dic6 [×2], C3×Q8 [×2], C22×S3, C22×C6, C42.C2, S3×C6 [×3], C2×Dic6, C4○D12 [×2], C6×Q8, C3×C4○D4 [×2], C3×Dic6 [×2], S3×C2×C6, C12.6Q8, C3×C42.C2, C6×Dic6, C3×C4○D12 [×2], C3×C12.6Q8

Smallest permutation representation of C3×C12.6Q8
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 29 33)(26 30 34)(27 31 35)(28 32 36)(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 65 69)(62 66 70)(63 67 71)(64 68 72)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(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 62 33 81)(2 63 34 82)(3 64 35 83)(4 65 36 84)(5 66 25 73)(6 67 26 74)(7 68 27 75)(8 69 28 76)(9 70 29 77)(10 71 30 78)(11 72 31 79)(12 61 32 80)(13 57 88 40)(14 58 89 41)(15 59 90 42)(16 60 91 43)(17 49 92 44)(18 50 93 45)(19 51 94 46)(20 52 95 47)(21 53 96 48)(22 54 85 37)(23 55 86 38)(24 56 87 39)
(1 95 27 14)(2 94 28 13)(3 93 29 24)(4 92 30 23)(5 91 31 22)(6 90 32 21)(7 89 33 20)(8 88 34 19)(9 87 35 18)(10 86 36 17)(11 85 25 16)(12 96 26 15)(37 72 60 73)(38 71 49 84)(39 70 50 83)(40 69 51 82)(41 68 52 81)(42 67 53 80)(43 66 54 79)(44 65 55 78)(45 64 56 77)(46 63 57 76)(47 62 58 75)(48 61 59 74)

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

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

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

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 Q8 D6 C4○D4 C3×S3 Dic6 C3×Q8 S3×C6 C4○D12 C3×C4○D4 C3×Dic6 C3×C4○D12 kernel C3×C12.6Q8 C3×Dic3⋊C4 C3×C4⋊Dic3 C122 C12.6Q8 Dic3⋊C4 C4⋊Dic3 C4×C12 C4×C12 C3×C12 C2×C12 C3×C6 C42 C12 C12 C2×C4 C6 C6 C4 C2 # reps 1 4 2 1 2 8 4 2 1 2 3 4 2 4 4 6 8 8 8 16

Matrix representation of C3×C12.6Q8 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 3 0 0 0 7 9
,
 1 0 0 0 0 12 0 0 0 0 5 0 0 0 10 8
,
 0 1 0 0 12 0 0 0 0 0 5 8 0 0 0 8
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[11,0,0,0,0,6,0,0,0,0,3,7,0,0,0,9],[1,0,0,0,0,12,0,0,0,0,5,10,0,0,0,8],[0,12,0,0,1,0,0,0,0,0,5,0,0,0,8,8] >;

C3×C12.6Q8 in GAP, Magma, Sage, TeX

C_3\times C_{12}._6Q_8
% in TeX

G:=Group("C3xC12.6Q8");
// GroupNames label

G:=SmallGroup(288,641);
// by ID

G=gap.SmallGroup(288,641);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,701,176,590,268,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^4=1,d^2=b^6*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=b^6*c^-1>;
// generators/relations

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