metabelian, supersoluble, monomial
Aliases: C6.6Dic12, C62.21D4, Dic6⋊1Dic3, (C3×C6).4Q16, C12.17(C4×S3), (C3×Dic6)⋊2C4, (C3×C12).36D4, (C2×C6).51D12, C4.2(S3×Dic3), C6.9(C24⋊C2), C6.37(D6⋊C4), (C2×C12).274D6, (C6×Dic6).3C2, (C2×Dic6).1S3, (C3×C6).11SD16, C6.1(C3⋊Q16), C3⋊1(Q8⋊2Dic3), C12.32(C3⋊D4), C2.8(D6⋊Dic3), (C6×C12).28C22, C6.2(Q8⋊2S3), C3⋊3(C2.Dic12), C12.23(C2×Dic3), C32⋊5(Q8⋊C4), C4.12(D6⋊S3), C12⋊Dic3.9C2, C6.7(C6.D4), C2.1(C32⋊3Q16), C2.2(C32⋊5SD16), C22.14(C3⋊D12), (C6×C3⋊C8).1C2, (C2×C4).54S32, (C2×C3⋊C8).1S3, (C3×C12).26(C2×C4), (C2×C6).32(C3⋊D4), (C3×C6).33(C22⋊C4), SmallGroup(288,214)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.Dic12
G = < a,b,c | a6=b24=1, c2=b12, bab-1=a-1, ac=ca, cbc-1=a3b-1 >
Subgroups: 330 in 97 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, Q8⋊C4, C3×Dic3, C3⋊Dic3, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C2×C24, C2×Dic6, C6×Q8, C3×C3⋊C8, C3×Dic6, C3×Dic6, C6×Dic3, C2×C3⋊Dic3, C6×C12, C2.Dic12, Q8⋊2Dic3, C6×C3⋊C8, C12⋊Dic3, C6×Dic6, C6.Dic12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, SD16, Q16, C4×S3, D12, C2×Dic3, C3⋊D4, Q8⋊C4, S32, C24⋊C2, Dic12, D6⋊C4, Q8⋊2S3, C3⋊Q16, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C2.Dic12, Q8⋊2Dic3, C32⋊5SD16, C32⋊3Q16, D6⋊Dic3, C6.Dic12
►On 96 points
(1 67 9 51 17 59)(2 60 18 52 10 68)(3 69 11 53 19 61)(4 62 20 54 12 70)(5 71 13 55 21 63)(6 64 22 56 14 72)(7 49 15 57 23 65)(8 66 24 58 16 50)(25 96 33 80 41 88)(26 89 42 81 34 73)(27 74 35 82 43 90)(28 91 44 83 36 75)(29 76 37 84 45 92)(30 93 46 85 38 77)(31 78 39 86 47 94)(32 95 48 87 40 79)
(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 85 13 73)(2 29 14 41)(3 83 15 95)(4 27 16 39)(5 81 17 93)(6 25 18 37)(7 79 19 91)(8 47 20 35)(9 77 21 89)(10 45 22 33)(11 75 23 87)(12 43 24 31)(26 67 38 55)(28 65 40 53)(30 63 42 51)(32 61 44 49)(34 59 46 71)(36 57 48 69)(50 86 62 74)(52 84 64 96)(54 82 66 94)(56 80 68 92)(58 78 70 90)(60 76 72 88)
G:=sub<Sym(96)| (1,67,9,51,17,59)(2,60,18,52,10,68)(3,69,11,53,19,61)(4,62,20,54,12,70)(5,71,13,55,21,63)(6,64,22,56,14,72)(7,49,15,57,23,65)(8,66,24,58,16,50)(25,96,33,80,41,88)(26,89,42,81,34,73)(27,74,35,82,43,90)(28,91,44,83,36,75)(29,76,37,84,45,92)(30,93,46,85,38,77)(31,78,39,86,47,94)(32,95,48,87,40,79), (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), (1,85,13,73)(2,29,14,41)(3,83,15,95)(4,27,16,39)(5,81,17,93)(6,25,18,37)(7,79,19,91)(8,47,20,35)(9,77,21,89)(10,45,22,33)(11,75,23,87)(12,43,24,31)(26,67,38,55)(28,65,40,53)(30,63,42,51)(32,61,44,49)(34,59,46,71)(36,57,48,69)(50,86,62,74)(52,84,64,96)(54,82,66,94)(56,80,68,92)(58,78,70,90)(60,76,72,88)>;
G:=Group( (1,67,9,51,17,59)(2,60,18,52,10,68)(3,69,11,53,19,61)(4,62,20,54,12,70)(5,71,13,55,21,63)(6,64,22,56,14,72)(7,49,15,57,23,65)(8,66,24,58,16,50)(25,96,33,80,41,88)(26,89,42,81,34,73)(27,74,35,82,43,90)(28,91,44,83,36,75)(29,76,37,84,45,92)(30,93,46,85,38,77)(31,78,39,86,47,94)(32,95,48,87,40,79), (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), (1,85,13,73)(2,29,14,41)(3,83,15,95)(4,27,16,39)(5,81,17,93)(6,25,18,37)(7,79,19,91)(8,47,20,35)(9,77,21,89)(10,45,22,33)(11,75,23,87)(12,43,24,31)(26,67,38,55)(28,65,40,53)(30,63,42,51)(32,61,44,49)(34,59,46,71)(36,57,48,69)(50,86,62,74)(52,84,64,96)(54,82,66,94)(56,80,68,92)(58,78,70,90)(60,76,72,88) );
G=PermutationGroup([[(1,67,9,51,17,59),(2,60,18,52,10,68),(3,69,11,53,19,61),(4,62,20,54,12,70),(5,71,13,55,21,63),(6,64,22,56,14,72),(7,49,15,57,23,65),(8,66,24,58,16,50),(25,96,33,80,41,88),(26,89,42,81,34,73),(27,74,35,82,43,90),(28,91,44,83,36,75),(29,76,37,84,45,92),(30,93,46,85,38,77),(31,78,39,86,47,94),(32,95,48,87,40,79)], [(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)], [(1,85,13,73),(2,29,14,41),(3,83,15,95),(4,27,16,39),(5,81,17,93),(6,25,18,37),(7,79,19,91),(8,47,20,35),(9,77,21,89),(10,45,22,33),(11,75,23,87),(12,43,24,31),(26,67,38,55),(28,65,40,53),(30,63,42,51),(32,61,44,49),(34,59,46,71),(36,57,48,69),(50,86,62,74),(52,84,64,96),(54,82,66,94),(56,80,68,92),(58,78,70,90),(60,76,72,88)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 12 | 12 | 36 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 6 | ··· | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | + | - | + | + | - | - | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | D4 | Dic3 | D6 | SD16 | Q16 | C4×S3 | C3⋊D4 | D12 | C3⋊D4 | C24⋊C2 | Dic12 | S32 | Q8⋊2S3 | C3⋊Q16 | S3×Dic3 | D6⋊S3 | C3⋊D12 | C32⋊5SD16 | C32⋊3Q16 |
| kernel | C6.Dic12 | C6×C3⋊C8 | C12⋊Dic3 | C6×Dic6 | C3×Dic6 | C2×C3⋊C8 | C2×Dic6 | C3×C12 | C62 | Dic6 | C2×C12 | C3×C6 | C3×C6 | C12 | C12 | C2×C6 | C2×C6 | C6 | C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C6.Dic12 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 59 | 55 | 0 | 0 | 0 | 0 |
| 8 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 7 |
| 0 | 0 | 0 | 0 | 66 | 59 |
| 24 | 25 | 0 | 0 | 0 | 0 |
| 47 | 49 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 14 |
| 0 | 0 | 0 | 0 | 7 | 66 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[59,8,0,0,0,0,55,26,0,0,0,0,0,0,0,27,0,0,0,0,27,0,0,0,0,0,0,0,66,66,0,0,0,0,7,59],[24,47,0,0,0,0,25,49,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,7,0,0,0,0,14,66] >;
C6.Dic12 in GAP, Magma, Sage, TeX
C_6.{\rm Dic}_{12} % in TeX
G:=Group("C6.Dic12"); // GroupNames label
G:=SmallGroup(288,214);
// by ID
G=gap.SmallGroup(288,214);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,92,422,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^6=b^24=1,c^2=b^12,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b^-1>;
// generators/relations