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G = C3×C12⋊Q8order 288 = 25·32

Direct product of C3 and C12⋊Q8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C12⋊Q8
G = < a,b,c,d | a3=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=c-1 >

Subgroups: 306 in 149 conjugacy classes, 74 normal (38 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×8], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C32, Dic3 [×4], Dic3 [×2], C12 [×4], C12 [×14], C2×C6 [×2], C2×C6, C42, C4⋊C4, C4⋊C4 [×3], C2×Q8 [×2], C3×C6 [×3], Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×7], C3×Q8 [×4], C4⋊Q8, C3×Dic3 [×4], C3×Dic3 [×2], C3×C12 [×2], C3×C12 [×2], C62, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×4], C2×Dic6 [×2], C6×Q8 [×2], C3×Dic6 [×4], C6×Dic3 [×2], C6×Dic3 [×2], C6×C12, C6×C12 [×2], C12⋊Q8, C3×C4⋊Q8, Dic3×C12, C3×Dic3⋊C4 [×2], C3×C4⋊Dic3, C32×C4⋊C4, C6×Dic6 [×2], C3×C12⋊Q8
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], Q8 [×4], C23, D6 [×3], C2×C6 [×7], C2×D4, C2×Q8 [×2], C3×S3, Dic6 [×2], C3×D4 [×2], C3×Q8 [×4], C22×S3, C22×C6, C4⋊Q8, S3×C6 [×3], C2×Dic6, S3×D4, S3×Q8, C6×D4, C6×Q8 [×2], C3×Dic6 [×2], S3×C2×C6, C12⋊Q8, C3×C4⋊Q8, C6×Dic6, C3×S3×D4, C3×S3×Q8, C3×C12⋊Q8

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6F 6G ··· 6O 12A 12B 12C 12D 12E ··· 12Z 12AA ··· 12AH 12AI 12AJ 12AK 12AL order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 1 1 2 2 2 2 2 4 4 6 6 6 6 12 12 1 ··· 1 2 ··· 2 2 2 2 2 4 ··· 4 6 ··· 6 12 12 12 12

72 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 2 2 4 4 4 4 type + + + + + + + + - - + - + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 Q8 Q8 D6 C3×S3 C3×D4 C3×Q8 Dic6 C3×Q8 S3×C6 C3×Dic6 S3×D4 S3×Q8 C3×S3×D4 C3×S3×Q8 kernel C3×C12⋊Q8 Dic3×C12 C3×Dic3⋊C4 C3×C4⋊Dic3 C32×C4⋊C4 C6×Dic6 C12⋊Q8 C4×Dic3 Dic3⋊C4 C4⋊Dic3 C3×C4⋊C4 C2×Dic6 C3×C4⋊C4 C3×Dic3 C3×Dic3 C3×C12 C2×C12 C4⋊C4 Dic3 Dic3 C12 C12 C2×C4 C4 C6 C6 C2 C2 # reps 1 1 2 1 1 2 2 2 4 2 2 4 1 2 2 2 3 2 4 4 4 4 6 8 1 1 2 2

Matrix representation of C3×C12⋊Q8 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 10 0 0 0 11 4 0 0 0 0 0 1 0 0 12 0
,
 5 0 0 0 1 8 0 0 0 0 5 0 0 0 0 8
,
 9 1 0 0 9 4 0 0 0 0 0 1 0 0 12 0
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[10,11,0,0,0,4,0,0,0,0,0,12,0,0,1,0],[5,1,0,0,0,8,0,0,0,0,5,0,0,0,0,8],[9,9,0,0,1,4,0,0,0,0,0,12,0,0,1,0] >;

C3×C12⋊Q8 in GAP, Magma, Sage, TeX

C_3\times C_{12}\rtimes Q_8
% in TeX

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

G:=SmallGroup(288,659);
// by ID

G=gap.SmallGroup(288,659);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,344,590,555,142,9414]);
// 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,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=c^-1>;
// generators/relations

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