direct product, metabelian, supersoluble, monomial
Aliases: C6×Dic3⋊C4, C62.16Q8, C62.118D4, C62.191C23, C6.6(C6×Q8), C6.38(C6×D4), (C6×Dic3)⋊9C4, C2.2(C6×Dic6), Dic3⋊4(C2×C12), (C2×Dic3)⋊4C12, (C2×C12).350D6, C62.80(C2×C4), C23.39(S3×C6), (C2×C6).23Dic6, C6.53(C2×Dic6), (C22×C12).16S3, C6.17(C22×C12), (C22×C12).18C6, C22.16(S3×C12), (C22×C6).171D6, C22.4(C3×Dic6), (C6×C12).282C22, (C2×C62).94C22, (C22×Dic3).5C6, (C6×Dic3).132C22, C6⋊1(C3×C4⋊C4), C3⋊2(C6×C4⋊C4), (C2×C6×C12).5C2, (C3×C6)⋊6(C4⋊C4), C32⋊12(C2×C4⋊C4), C2.18(S3×C2×C12), C6.117(S3×C2×C4), C2.1(C6×C3⋊D4), (C2×C6).8(C3×Q8), (C2×C4).67(S3×C6), (C2×C6).86(C4×S3), (C2×C6).44(C3×D4), C22.20(S3×C2×C6), (C3×C6).49(C2×Q8), (C2×C12).90(C2×C6), (C2×C6).21(C2×C12), (C3×C6).250(C2×D4), C6.139(C2×C3⋊D4), (C22×C4).7(C3×S3), (Dic3×C2×C6).12C2, (C3×Dic3)⋊17(C2×C4), (C3×C6).89(C22×C4), (C22×C6).58(C2×C6), (C2×C6).46(C22×C6), C22.19(C3×C3⋊D4), (C2×C6).112(C3⋊D4), (C2×C6).324(C22×S3), (C2×Dic3).32(C2×C6), SmallGroup(288,694)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×Dic3⋊C4
G = < a,b,c,d | a6=b6=d4=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >
Subgroups: 378 in 211 conjugacy classes, 114 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C2×C4⋊C4, C3×Dic3, C3×Dic3, C3×C12, C62, C62, Dic3⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C22×C12, C6×Dic3, C6×Dic3, C6×C12, C6×C12, C2×C62, C2×Dic3⋊C4, C6×C4⋊C4, C3×Dic3⋊C4, Dic3×C2×C6, C2×C6×C12, C6×Dic3⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C23, C12, D6, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C2×C4⋊C4, S3×C6, Dic3⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C22×C12, C6×D4, C6×Q8, C3×Dic6, S3×C12, C3×C3⋊D4, S3×C2×C6, C2×Dic3⋊C4, C6×C4⋊C4, C3×Dic3⋊C4, C6×Dic6, S3×C2×C12, C6×C3⋊D4, C6×Dic3⋊C4
►On 96 points
(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 7 5 11 3 9)(2 8 6 12 4 10)(13 54 17 52 15 50)(14 49 18 53 16 51)(19 35 23 33 21 31)(20 36 24 34 22 32)(25 64 27 66 29 62)(26 65 28 61 30 63)(37 43 41 47 39 45)(38 44 42 48 40 46)(55 68 57 70 59 72)(56 69 58 71 60 67)(73 86 75 88 77 90)(74 87 76 89 78 85)(79 92 81 94 83 96)(80 93 82 95 84 91)
(1 58 11 67)(2 59 12 68)(3 60 7 69)(4 55 8 70)(5 56 9 71)(6 57 10 72)(13 92 52 83)(14 93 53 84)(15 94 54 79)(16 95 49 80)(17 96 50 81)(18 91 51 82)(19 74 33 89)(20 75 34 90)(21 76 35 85)(22 77 36 86)(23 78 31 87)(24 73 32 88)(25 43 66 39)(26 44 61 40)(27 45 62 41)(28 46 63 42)(29 47 64 37)(30 48 65 38)
(1 13 19 40)(2 14 20 41)(3 15 21 42)(4 16 22 37)(5 17 23 38)(6 18 24 39)(7 54 35 46)(8 49 36 47)(9 50 31 48)(10 51 32 43)(11 52 33 44)(12 53 34 45)(25 72 91 88)(26 67 92 89)(27 68 93 90)(28 69 94 85)(29 70 95 86)(30 71 96 87)(55 80 77 64)(56 81 78 65)(57 82 73 66)(58 83 74 61)(59 84 75 62)(60 79 76 63)
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), (1,7,5,11,3,9)(2,8,6,12,4,10)(13,54,17,52,15,50)(14,49,18,53,16,51)(19,35,23,33,21,31)(20,36,24,34,22,32)(25,64,27,66,29,62)(26,65,28,61,30,63)(37,43,41,47,39,45)(38,44,42,48,40,46)(55,68,57,70,59,72)(56,69,58,71,60,67)(73,86,75,88,77,90)(74,87,76,89,78,85)(79,92,81,94,83,96)(80,93,82,95,84,91), (1,58,11,67)(2,59,12,68)(3,60,7,69)(4,55,8,70)(5,56,9,71)(6,57,10,72)(13,92,52,83)(14,93,53,84)(15,94,54,79)(16,95,49,80)(17,96,50,81)(18,91,51,82)(19,74,33,89)(20,75,34,90)(21,76,35,85)(22,77,36,86)(23,78,31,87)(24,73,32,88)(25,43,66,39)(26,44,61,40)(27,45,62,41)(28,46,63,42)(29,47,64,37)(30,48,65,38), (1,13,19,40)(2,14,20,41)(3,15,21,42)(4,16,22,37)(5,17,23,38)(6,18,24,39)(7,54,35,46)(8,49,36,47)(9,50,31,48)(10,51,32,43)(11,52,33,44)(12,53,34,45)(25,72,91,88)(26,67,92,89)(27,68,93,90)(28,69,94,85)(29,70,95,86)(30,71,96,87)(55,80,77,64)(56,81,78,65)(57,82,73,66)(58,83,74,61)(59,84,75,62)(60,79,76,63)>;
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), (1,7,5,11,3,9)(2,8,6,12,4,10)(13,54,17,52,15,50)(14,49,18,53,16,51)(19,35,23,33,21,31)(20,36,24,34,22,32)(25,64,27,66,29,62)(26,65,28,61,30,63)(37,43,41,47,39,45)(38,44,42,48,40,46)(55,68,57,70,59,72)(56,69,58,71,60,67)(73,86,75,88,77,90)(74,87,76,89,78,85)(79,92,81,94,83,96)(80,93,82,95,84,91), (1,58,11,67)(2,59,12,68)(3,60,7,69)(4,55,8,70)(5,56,9,71)(6,57,10,72)(13,92,52,83)(14,93,53,84)(15,94,54,79)(16,95,49,80)(17,96,50,81)(18,91,51,82)(19,74,33,89)(20,75,34,90)(21,76,35,85)(22,77,36,86)(23,78,31,87)(24,73,32,88)(25,43,66,39)(26,44,61,40)(27,45,62,41)(28,46,63,42)(29,47,64,37)(30,48,65,38), (1,13,19,40)(2,14,20,41)(3,15,21,42)(4,16,22,37)(5,17,23,38)(6,18,24,39)(7,54,35,46)(8,49,36,47)(9,50,31,48)(10,51,32,43)(11,52,33,44)(12,53,34,45)(25,72,91,88)(26,67,92,89)(27,68,93,90)(28,69,94,85)(29,70,95,86)(30,71,96,87)(55,80,77,64)(56,81,78,65)(57,82,73,66)(58,83,74,61)(59,84,75,62)(60,79,76,63) );
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)], [(1,7,5,11,3,9),(2,8,6,12,4,10),(13,54,17,52,15,50),(14,49,18,53,16,51),(19,35,23,33,21,31),(20,36,24,34,22,32),(25,64,27,66,29,62),(26,65,28,61,30,63),(37,43,41,47,39,45),(38,44,42,48,40,46),(55,68,57,70,59,72),(56,69,58,71,60,67),(73,86,75,88,77,90),(74,87,76,89,78,85),(79,92,81,94,83,96),(80,93,82,95,84,91)], [(1,58,11,67),(2,59,12,68),(3,60,7,69),(4,55,8,70),(5,56,9,71),(6,57,10,72),(13,92,52,83),(14,93,53,84),(15,94,54,79),(16,95,49,80),(17,96,50,81),(18,91,51,82),(19,74,33,89),(20,75,34,90),(21,76,35,85),(22,77,36,86),(23,78,31,87),(24,73,32,88),(25,43,66,39),(26,44,61,40),(27,45,62,41),(28,46,63,42),(29,47,64,37),(30,48,65,38)], [(1,13,19,40),(2,14,20,41),(3,15,21,42),(4,16,22,37),(5,17,23,38),(6,18,24,39),(7,54,35,46),(8,49,36,47),(9,50,31,48),(10,51,32,43),(11,52,33,44),(12,53,34,45),(25,72,91,88),(26,67,92,89),(27,68,93,90),(28,69,94,85),(29,70,95,86),(30,71,96,87),(55,80,77,64),(56,81,78,65),(57,82,73,66),(58,83,74,61),(59,84,75,62),(60,79,76,63)]])
108 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6N | 6O | ··· | 6AI | 12A | ··· | 12AF | 12AG | ··· | 12AV |
| order | 1 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D4 | Q8 | D6 | D6 | C3×S3 | Dic6 | C4×S3 | C3⋊D4 | C3×D4 | C3×Q8 | S3×C6 | S3×C6 | C3×Dic6 | S3×C12 | C3×C3⋊D4 |
| kernel | C6×Dic3⋊C4 | C3×Dic3⋊C4 | Dic3×C2×C6 | C2×C6×C12 | C2×Dic3⋊C4 | C6×Dic3 | Dic3⋊C4 | C22×Dic3 | C22×C12 | C2×Dic3 | C22×C12 | C62 | C62 | C2×C12 | C22×C6 | C22×C4 | C2×C6 | C2×C6 | C2×C6 | C2×C6 | C2×C6 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 8 | 4 | 2 | 16 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 8 | 8 | 8 |
Matrix representation of C6×Dic3⋊C4 ►in GL4(𝔽13) generated by
| 10 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 3 |
| 0 | 0 | 8 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [10,0,0,0,0,9,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,9,10],[1,0,0,0,0,1,0,0,0,0,12,8,0,0,3,1],[1,0,0,0,0,5,0,0,0,0,8,0,0,0,2,5] >;
C6×Dic3⋊C4 in GAP, Magma, Sage, TeX
C_6\times {\rm Dic}_3\rtimes C_4 % in TeX
G:=Group("C6xDic3:C4"); // GroupNames label
G:=SmallGroup(288,694);
// by ID
G=gap.SmallGroup(288,694);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,1094,142,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=d^4=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations