metabelian, supersoluble, monomial
Aliases: C12.5Dic6, C62.26D4, C3⋊C8⋊1Dic3, (C3×C12).5Q8, C12.89(C4×S3), (C2×C6).54D12, C3⋊1(C8⋊Dic3), (C2×C12).80D6, C4⋊Dic3.2S3, C32⋊5(C4.Q8), C6.3(D4.S3), C6.2(C4⋊Dic3), C4.11(S3×Dic3), (C3×C6).13SD16, C6.12(C24⋊C2), C6.3(Dic3⋊C4), (C6×C12).33C22, C6.3(Q8⋊2S3), C3⋊1(C12.Q8), C12.10(C2×Dic3), C4.1(C32⋊2Q8), C2.3(Dic3⋊Dic3), C2.3(C32⋊5SD16), C12⋊Dic3.10C2, C2.3(D12.S3), C22.16(C3⋊D12), (C3×C3⋊C8)⋊2C4, (C2×C4).58S32, (C2×C3⋊C8).3S3, (C6×C3⋊C8).12C2, (C3×C6).20(C4⋊C4), (C3×C12).31(C2×C4), (C3×C4⋊Dic3).3C2, (C2×C6).34(C3⋊D4), SmallGroup(288,221)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.Dic6
G = < a,b,c | a12=1, b12=a6, c2=a9b6, bab-1=a5, cac-1=a7, cbc-1=b11 >
Subgroups: 298 in 85 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C4.Q8, C3×Dic3, C3⋊Dic3, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C2×C24, C3×C3⋊C8, C6×Dic3, C2×C3⋊Dic3, C6×C12, C12.Q8, C8⋊Dic3, C6×C3⋊C8, C3×C4⋊Dic3, C12⋊Dic3, C12.Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, SD16, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C4.Q8, S32, C24⋊C2, Dic3⋊C4, C4⋊Dic3, D4.S3, Q8⋊2S3, S3×Dic3, C3⋊D12, C32⋊2Q8, C12.Q8, C8⋊Dic3, D12.S3, C32⋊5SD16, Dic3⋊Dic3, C12.Dic6
►On 96 points
(1 84 5 88 9 92 13 96 17 76 21 80)(2 93 22 89 18 85 14 81 10 77 6 73)(3 86 7 90 11 94 15 74 19 78 23 82)(4 95 24 91 20 87 16 83 12 79 8 75)(25 71 29 51 33 55 37 59 41 63 45 67)(26 56 46 52 42 72 38 68 34 64 30 60)(27 49 31 53 35 57 39 61 43 65 47 69)(28 58 48 54 44 50 40 70 36 66 32 62)
(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 82 30)(2 61 83 41)(3 72 84 28)(4 59 85 39)(5 70 86 26)(6 57 87 37)(7 68 88 48)(8 55 89 35)(9 66 90 46)(10 53 91 33)(11 64 92 44)(12 51 93 31)(13 62 94 42)(14 49 95 29)(15 60 96 40)(16 71 73 27)(17 58 74 38)(18 69 75 25)(19 56 76 36)(20 67 77 47)(21 54 78 34)(22 65 79 45)(23 52 80 32)(24 63 81 43)
G:=sub<Sym(96)| (1,84,5,88,9,92,13,96,17,76,21,80)(2,93,22,89,18,85,14,81,10,77,6,73)(3,86,7,90,11,94,15,74,19,78,23,82)(4,95,24,91,20,87,16,83,12,79,8,75)(25,71,29,51,33,55,37,59,41,63,45,67)(26,56,46,52,42,72,38,68,34,64,30,60)(27,49,31,53,35,57,39,61,43,65,47,69)(28,58,48,54,44,50,40,70,36,66,32,62), (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), (1,50,82,30)(2,61,83,41)(3,72,84,28)(4,59,85,39)(5,70,86,26)(6,57,87,37)(7,68,88,48)(8,55,89,35)(9,66,90,46)(10,53,91,33)(11,64,92,44)(12,51,93,31)(13,62,94,42)(14,49,95,29)(15,60,96,40)(16,71,73,27)(17,58,74,38)(18,69,75,25)(19,56,76,36)(20,67,77,47)(21,54,78,34)(22,65,79,45)(23,52,80,32)(24,63,81,43)>;
G:=Group( (1,84,5,88,9,92,13,96,17,76,21,80)(2,93,22,89,18,85,14,81,10,77,6,73)(3,86,7,90,11,94,15,74,19,78,23,82)(4,95,24,91,20,87,16,83,12,79,8,75)(25,71,29,51,33,55,37,59,41,63,45,67)(26,56,46,52,42,72,38,68,34,64,30,60)(27,49,31,53,35,57,39,61,43,65,47,69)(28,58,48,54,44,50,40,70,36,66,32,62), (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), (1,50,82,30)(2,61,83,41)(3,72,84,28)(4,59,85,39)(5,70,86,26)(6,57,87,37)(7,68,88,48)(8,55,89,35)(9,66,90,46)(10,53,91,33)(11,64,92,44)(12,51,93,31)(13,62,94,42)(14,49,95,29)(15,60,96,40)(16,71,73,27)(17,58,74,38)(18,69,75,25)(19,56,76,36)(20,67,77,47)(21,54,78,34)(22,65,79,45)(23,52,80,32)(24,63,81,43) );
G=PermutationGroup([[(1,84,5,88,9,92,13,96,17,76,21,80),(2,93,22,89,18,85,14,81,10,77,6,73),(3,86,7,90,11,94,15,74,19,78,23,82),(4,95,24,91,20,87,16,83,12,79,8,75),(25,71,29,51,33,55,37,59,41,63,45,67),(26,56,46,52,42,72,38,68,34,64,30,60),(27,49,31,53,35,57,39,61,43,65,47,69),(28,58,48,54,44,50,40,70,36,66,32,62)], [(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)], [(1,50,82,30),(2,61,83,41),(3,72,84,28),(4,59,85,39),(5,70,86,26),(6,57,87,37),(7,68,88,48),(8,55,89,35),(9,66,90,46),(10,53,91,33),(11,64,92,44),(12,51,93,31),(13,62,94,42),(14,49,95,29),(15,60,96,40),(16,71,73,27),(17,58,74,38),(18,69,75,25),(19,56,76,36),(20,67,77,47),(21,54,78,34),(22,65,79,45),(23,52,80,32),(24,63,81,43)]])
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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | + | + | - | + | - | - | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | Q8 | D4 | Dic3 | D6 | SD16 | Dic6 | C4×S3 | D12 | C3⋊D4 | C24⋊C2 | S32 | D4.S3 | Q8⋊2S3 | S3×Dic3 | C32⋊2Q8 | C3⋊D12 | D12.S3 | C32⋊5SD16 |
| kernel | C12.Dic6 | C6×C3⋊C8 | C3×C4⋊Dic3 | C12⋊Dic3 | C3×C3⋊C8 | C2×C3⋊C8 | C4⋊Dic3 | C3×C12 | C62 | C3⋊C8 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C2×C6 | C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C12.Dic6 ►in GL6(𝔽73)
| 72 | 2 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 67 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 7 |
| 0 | 0 | 0 | 0 | 66 | 66 |
| 46 | 54 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 5 |
| 0 | 0 | 0 | 0 | 19 | 59 |
G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,2,1,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,67,0,0,0,0,12,12,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,59,66,0,0,0,0,7,66],[46,0,0,0,0,0,54,27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,14,19,0,0,0,0,5,59] >;
C12.Dic6 in GAP, Magma, Sage, TeX
C_{12}.{\rm Dic}_6 % in TeX
G:=Group("C12.Dic6"); // GroupNames label
G:=SmallGroup(288,221);
// by ID
G=gap.SmallGroup(288,221);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,85,36,422,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9*b^6,b*a*b^-1=a^5,c*a*c^-1=a^7,c*b*c^-1=b^11>;
// generators/relations