direct product, metabelian, supersoluble, monomial
Aliases: C3×C12⋊Q8, C12⋊5Dic6, C62.179C23, C12⋊(C3×Q8), (C3×C12)⋊5Q8, C4⋊1(C3×Dic6), C6.22(C6×D4), C6.52(S3×Q8), C6.10(C6×Q8), Dic3⋊1(C3×Q8), (C3×Dic3)⋊8Q8, C6.181(S3×D4), C32⋊11(C4⋊Q8), C2.7(C6×Dic6), (C2×C12).233D6, Dic3⋊C4.2C6, C4⋊Dic3.11C6, (C2×Dic6).3C6, Dic3.2(C3×D4), (C4×Dic3).1C6, C6.51(C2×Dic6), (C3×Dic3).29D4, (C6×Dic6).18C2, (C6×C12).112C22, (Dic3×C12).10C2, (C6×Dic3).95C22, C3⋊2(C3×C4⋊Q8), C2.4(C3×S3×Q8), C2.11(C3×S3×D4), (C3×C4⋊C4).5C6, C4⋊C4.4(C3×S3), (C2×C4).8(S3×C6), (C3×C4⋊C4).27S3, (C2×C12).4(C2×C6), C22.46(S3×C2×C6), (C3×C6).62(C2×Q8), (C3×C6).210(C2×D4), (C32×C4⋊C4).8C2, (C3×C4⋊Dic3).18C2, (C2×C6).34(C22×C6), (C3×Dic3⋊C4).10C2, (C2×C6).312(C22×S3), (C2×Dic3).25(C2×C6), SmallGroup(288,659)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C32, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C4⋊C4, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C4⋊Q8, C3×Dic3, C3×Dic3, C3×C12, C3×C12, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, C3×Dic6, C6×Dic3, C6×Dic3, C6×C12, C6×C12, C12⋊Q8, C3×C4⋊Q8, Dic3×C12, C3×Dic3⋊C4, C3×C4⋊Dic3, C32×C4⋊C4, C6×Dic6, C3×C12⋊Q8
Quotients: C1, C2, C3, C22, S3, C6, D4, Q8, C23, D6, C2×C6, C2×D4, C2×Q8, C3×S3, Dic6, C3×D4, C3×Q8, C22×S3, C22×C6, C4⋊Q8, S3×C6, C2×Dic6, S3×D4, S3×Q8, C6×D4, C6×Q8, C3×Dic6, S3×C2×C6, C12⋊Q8, C3×C4⋊Q8, C6×Dic6, C3×S3×D4, C3×S3×Q8, C3×C12⋊Q8
►On 96 points
(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 41 45)(38 42 46)(39 43 47)(40 44 48)(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 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 77 35 46)(2 84 36 41)(3 79 25 48)(4 74 26 43)(5 81 27 38)(6 76 28 45)(7 83 29 40)(8 78 30 47)(9 73 31 42)(10 80 32 37)(11 75 33 44)(12 82 34 39)(13 57 63 93)(14 52 64 88)(15 59 65 95)(16 54 66 90)(17 49 67 85)(18 56 68 92)(19 51 69 87)(20 58 70 94)(21 53 71 89)(22 60 72 96)(23 55 61 91)(24 50 62 86)
(1 71 35 21)(2 64 36 14)(3 69 25 19)(4 62 26 24)(5 67 27 17)(6 72 28 22)(7 65 29 15)(8 70 30 20)(9 63 31 13)(10 68 32 18)(11 61 33 23)(12 66 34 16)(37 92 80 56)(38 85 81 49)(39 90 82 54)(40 95 83 59)(41 88 84 52)(42 93 73 57)(43 86 74 50)(44 91 75 55)(45 96 76 60)(46 89 77 53)(47 94 78 58)(48 87 79 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)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(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)(85,93,89)(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), (1,77,35,46)(2,84,36,41)(3,79,25,48)(4,74,26,43)(5,81,27,38)(6,76,28,45)(7,83,29,40)(8,78,30,47)(9,73,31,42)(10,80,32,37)(11,75,33,44)(12,82,34,39)(13,57,63,93)(14,52,64,88)(15,59,65,95)(16,54,66,90)(17,49,67,85)(18,56,68,92)(19,51,69,87)(20,58,70,94)(21,53,71,89)(22,60,72,96)(23,55,61,91)(24,50,62,86), (1,71,35,21)(2,64,36,14)(3,69,25,19)(4,62,26,24)(5,67,27,17)(6,72,28,22)(7,65,29,15)(8,70,30,20)(9,63,31,13)(10,68,32,18)(11,61,33,23)(12,66,34,16)(37,92,80,56)(38,85,81,49)(39,90,82,54)(40,95,83,59)(41,88,84,52)(42,93,73,57)(43,86,74,50)(44,91,75,55)(45,96,76,60)(46,89,77,53)(47,94,78,58)(48,87,79,51)>;
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)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(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)(85,93,89)(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), (1,77,35,46)(2,84,36,41)(3,79,25,48)(4,74,26,43)(5,81,27,38)(6,76,28,45)(7,83,29,40)(8,78,30,47)(9,73,31,42)(10,80,32,37)(11,75,33,44)(12,82,34,39)(13,57,63,93)(14,52,64,88)(15,59,65,95)(16,54,66,90)(17,49,67,85)(18,56,68,92)(19,51,69,87)(20,58,70,94)(21,53,71,89)(22,60,72,96)(23,55,61,91)(24,50,62,86), (1,71,35,21)(2,64,36,14)(3,69,25,19)(4,62,26,24)(5,67,27,17)(6,72,28,22)(7,65,29,15)(8,70,30,20)(9,63,31,13)(10,68,32,18)(11,61,33,23)(12,66,34,16)(37,92,80,56)(38,85,81,49)(39,90,82,54)(40,95,83,59)(41,88,84,52)(42,93,73,57)(43,86,74,50)(44,91,75,55)(45,96,76,60)(46,89,77,53)(47,94,78,58)(48,87,79,51) );
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),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(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),(85,93,89),(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)], [(1,77,35,46),(2,84,36,41),(3,79,25,48),(4,74,26,43),(5,81,27,38),(6,76,28,45),(7,83,29,40),(8,78,30,47),(9,73,31,42),(10,80,32,37),(11,75,33,44),(12,82,34,39),(13,57,63,93),(14,52,64,88),(15,59,65,95),(16,54,66,90),(17,49,67,85),(18,56,68,92),(19,51,69,87),(20,58,70,94),(21,53,71,89),(22,60,72,96),(23,55,61,91),(24,50,62,86)], [(1,71,35,21),(2,64,36,14),(3,69,25,19),(4,62,26,24),(5,67,27,17),(6,72,28,22),(7,65,29,15),(8,70,30,20),(9,63,31,13),(10,68,32,18),(11,61,33,23),(12,66,34,16),(37,92,80,56),(38,85,81,49),(39,90,82,54),(40,95,83,59),(41,88,84,52),(42,93,73,57),(43,86,74,50),(44,91,75,55),(45,96,76,60),(46,89,77,53),(47,94,78,58),(48,87,79,51)]])
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