metabelian, supersoluble, monomial
Aliases: C12.27D12, C62.30C23, D6⋊Dic3⋊2C2, (C4×Dic3)⋊3S3, (C2×D12).3S3, (C6×D12).8C2, C6.75(C2×D12), (C3×C12).75D4, (Dic3×C12)⋊8C2, (C2×C12).130D6, (C22×S3).6D6, C3⋊4(C42⋊7S3), C6.41(C4○D12), C4.9(C3⋊D12), C12.74(C3⋊D4), (C6×C12).92C22, (C2×Dic3).94D6, C32⋊3(C4.4D4), C6.17(D4⋊2S3), C3⋊1(C23.12D6), C2.10(D12⋊5S3), (C6×Dic3).107C22, (C2×C4).74S32, C22.87(C2×S32), (C3×C6).83(C2×D4), C6.11(C2×C3⋊D4), (S3×C2×C6).6C22, (C3×C6).17(C4○D4), C2.15(C2×C3⋊D12), (C2×C6).49(C22×S3), (C2×C32⋊4Q8)⋊12C2, (C2×C3⋊Dic3).28C22, SmallGroup(288,508)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.27D12
G = < a,b,c | a12=b12=1, c2=a6, bab-1=a5, cac-1=a-1, cbc-1=a6b-1 >
Subgroups: 642 in 171 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C3×S3, C3×C6, C3×C6, Dic6, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4.4D4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C62, C4×Dic3, D6⋊C4, C6.D4, C4×C12, C2×Dic6, C2×D12, C6×D4, C3×D12, C6×Dic3, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C42⋊7S3, C23.12D6, D6⋊Dic3, Dic3×C12, C6×D12, C2×C32⋊4Q8, C12.27D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4.4D4, S32, C2×D12, C4○D12, D4⋊2S3, C2×C3⋊D4, C3⋊D12, C2×S32, C42⋊7S3, C23.12D6, D12⋊5S3, C2×C3⋊D12, C12.27D12
►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 47 89 14 5 43 93 22 9 39 85 18)(2 40 90 19 6 48 94 15 10 44 86 23)(3 45 91 24 7 41 95 20 11 37 87 16)(4 38 92 17 8 46 96 13 12 42 88 21)(25 63 57 74 29 71 49 82 33 67 53 78)(26 68 58 79 30 64 50 75 34 72 54 83)(27 61 59 84 31 69 51 80 35 65 55 76)(28 66 60 77 32 62 52 73 36 70 56 81)
(1 35 7 29)(2 34 8 28)(3 33 9 27)(4 32 10 26)(5 31 11 25)(6 30 12 36)(13 64 19 70)(14 63 20 69)(15 62 21 68)(16 61 22 67)(17 72 23 66)(18 71 24 65)(37 84 43 78)(38 83 44 77)(39 82 45 76)(40 81 46 75)(41 80 47 74)(42 79 48 73)(49 85 55 91)(50 96 56 90)(51 95 57 89)(52 94 58 88)(53 93 59 87)(54 92 60 86)
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), (1,47,89,14,5,43,93,22,9,39,85,18)(2,40,90,19,6,48,94,15,10,44,86,23)(3,45,91,24,7,41,95,20,11,37,87,16)(4,38,92,17,8,46,96,13,12,42,88,21)(25,63,57,74,29,71,49,82,33,67,53,78)(26,68,58,79,30,64,50,75,34,72,54,83)(27,61,59,84,31,69,51,80,35,65,55,76)(28,66,60,77,32,62,52,73,36,70,56,81), (1,35,7,29)(2,34,8,28)(3,33,9,27)(4,32,10,26)(5,31,11,25)(6,30,12,36)(13,64,19,70)(14,63,20,69)(15,62,21,68)(16,61,22,67)(17,72,23,66)(18,71,24,65)(37,84,43,78)(38,83,44,77)(39,82,45,76)(40,81,46,75)(41,80,47,74)(42,79,48,73)(49,85,55,91)(50,96,56,90)(51,95,57,89)(52,94,58,88)(53,93,59,87)(54,92,60,86)>;
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), (1,47,89,14,5,43,93,22,9,39,85,18)(2,40,90,19,6,48,94,15,10,44,86,23)(3,45,91,24,7,41,95,20,11,37,87,16)(4,38,92,17,8,46,96,13,12,42,88,21)(25,63,57,74,29,71,49,82,33,67,53,78)(26,68,58,79,30,64,50,75,34,72,54,83)(27,61,59,84,31,69,51,80,35,65,55,76)(28,66,60,77,32,62,52,73,36,70,56,81), (1,35,7,29)(2,34,8,28)(3,33,9,27)(4,32,10,26)(5,31,11,25)(6,30,12,36)(13,64,19,70)(14,63,20,69)(15,62,21,68)(16,61,22,67)(17,72,23,66)(18,71,24,65)(37,84,43,78)(38,83,44,77)(39,82,45,76)(40,81,46,75)(41,80,47,74)(42,79,48,73)(49,85,55,91)(50,96,56,90)(51,95,57,89)(52,94,58,88)(53,93,59,87)(54,92,60,86) );
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)], [(1,47,89,14,5,43,93,22,9,39,85,18),(2,40,90,19,6,48,94,15,10,44,86,23),(3,45,91,24,7,41,95,20,11,37,87,16),(4,38,92,17,8,46,96,13,12,42,88,21),(25,63,57,74,29,71,49,82,33,67,53,78),(26,68,58,79,30,64,50,75,34,72,54,83),(27,61,59,84,31,69,51,80,35,65,55,76),(28,66,60,77,32,62,52,73,36,70,56,81)], [(1,35,7,29),(2,34,8,28),(3,33,9,27),(4,32,10,26),(5,31,11,25),(6,30,12,36),(13,64,19,70),(14,63,20,69),(15,62,21,68),(16,61,22,67),(17,72,23,66),(18,71,24,65),(37,84,43,78),(38,83,44,77),(39,82,45,76),(40,81,46,75),(41,80,47,74),(42,79,48,73),(49,85,55,91),(50,96,56,90),(51,95,57,89),(52,94,58,88),(53,93,59,87),(54,92,60,86)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | ··· | 12R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 6 | 6 | 36 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D6 | C4○D4 | D12 | C3⋊D4 | C4○D12 | S32 | D4⋊2S3 | C3⋊D12 | C2×S32 | D12⋊5S3 |
| kernel | C12.27D12 | D6⋊Dic3 | Dic3×C12 | C6×D12 | C2×C32⋊4Q8 | C4×Dic3 | C2×D12 | C3×C12 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | C12 | C12 | C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 2 | 2 | 1 | 4 |
Matrix representation of C12.27D12 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 6 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 5 | 2 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 2 | 11 |
| 9 | 9 | 0 | 0 | 0 | 0 |
| 7 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 2 | 11 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,3,7,0,0,0,0,6,10],[5,0,0,0,0,0,2,8,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,9,2,0,0,0,0,11,11],[9,7,0,0,0,0,9,4,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,2,2,0,0,0,0,4,11] >;
C12.27D12 in GAP, Magma, Sage, TeX
C_{12}._{27}D_{12} % in TeX
G:=Group("C12.27D12"); // GroupNames label
G:=SmallGroup(288,508);
// by ID
G=gap.SmallGroup(288,508);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,176,422,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=1,c^2=a^6,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;
// generators/relations