metabelian, supersoluble, monomial
Aliases: C12.15Dic6, C6.5(S3×C8), C32⋊7(C4⋊C8), C3⋊Dic3⋊4C8, (C3×C12).18Q8, C6.3(C8⋊S3), C3⋊1(Dic3⋊C8), (C3×C12).113D4, (C2×C12).298D6, C62.32(C2×C4), (C3×C6).8M4(2), C12.96(C3⋊D4), C6.2(Dic3⋊C4), C4.8(C32⋊2Q8), C4.18(D6⋊S3), (C6×C12).203C22, C2.3(C12.31D6), C2.5(C12.29D6), C2.1(C62.C22), C22.10(C6.D6), (C6×C3⋊C8).4C2, (C2×C4).131S32, (C2×C3⋊C8).10S3, (C3×C6).21(C2×C8), (C2×C6).28(C4×S3), (C3×C6).19(C4⋊C4), (C2×C3⋊Dic3).9C4, (C4×C3⋊Dic3).11C2, SmallGroup(288,220)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.15Dic6
G = < a,b,c | a12=1, b12=a6, c2=a6b6, bab-1=a5, ac=ca, cbc-1=a3b11 >
Subgroups: 266 in 91 conjugacy classes, 42 normal (20 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C2×C8, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C4⋊C8, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C2×C3⋊C8, C4×Dic3, C2×C24, C3×C3⋊C8, C2×C3⋊Dic3, C6×C12, Dic3⋊C8, C6×C3⋊C8, C4×C3⋊Dic3, C12.15Dic6
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, D6, C4⋊C4, C2×C8, M4(2), Dic6, C4×S3, C3⋊D4, C4⋊C8, S32, S3×C8, C8⋊S3, Dic3⋊C4, C6.D6, D6⋊S3, C32⋊2Q8, Dic3⋊C8, C12.29D6, C12.31D6, C62.C22, C12.15Dic6
►On 96 points
(1 26 5 30 9 34 13 38 17 42 21 46)(2 35 22 31 18 27 14 47 10 43 6 39)(3 28 7 32 11 36 15 40 19 44 23 48)(4 37 24 33 20 29 16 25 12 45 8 41)(49 74 69 94 65 90 61 86 57 82 53 78)(50 91 54 95 58 75 62 79 66 83 70 87)(51 76 71 96 67 92 63 88 59 84 55 80)(52 93 56 73 60 77 64 81 68 85 72 89)
(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 88 19 82 13 76 7 94)(2 66 20 60 14 54 8 72)(3 74 21 92 15 86 9 80)(4 52 22 70 16 64 10 58)(5 84 23 78 17 96 11 90)(6 62 24 56 18 50 12 68)(25 81 43 75 37 93 31 87)(26 59 44 53 38 71 32 65)(27 91 45 85 39 79 33 73)(28 69 46 63 40 57 34 51)(29 77 47 95 41 89 35 83)(30 55 48 49 42 67 36 61)
G:=sub<Sym(96)| (1,26,5,30,9,34,13,38,17,42,21,46)(2,35,22,31,18,27,14,47,10,43,6,39)(3,28,7,32,11,36,15,40,19,44,23,48)(4,37,24,33,20,29,16,25,12,45,8,41)(49,74,69,94,65,90,61,86,57,82,53,78)(50,91,54,95,58,75,62,79,66,83,70,87)(51,76,71,96,67,92,63,88,59,84,55,80)(52,93,56,73,60,77,64,81,68,85,72,89), (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), (1,88,19,82,13,76,7,94)(2,66,20,60,14,54,8,72)(3,74,21,92,15,86,9,80)(4,52,22,70,16,64,10,58)(5,84,23,78,17,96,11,90)(6,62,24,56,18,50,12,68)(25,81,43,75,37,93,31,87)(26,59,44,53,38,71,32,65)(27,91,45,85,39,79,33,73)(28,69,46,63,40,57,34,51)(29,77,47,95,41,89,35,83)(30,55,48,49,42,67,36,61)>;
G:=Group( (1,26,5,30,9,34,13,38,17,42,21,46)(2,35,22,31,18,27,14,47,10,43,6,39)(3,28,7,32,11,36,15,40,19,44,23,48)(4,37,24,33,20,29,16,25,12,45,8,41)(49,74,69,94,65,90,61,86,57,82,53,78)(50,91,54,95,58,75,62,79,66,83,70,87)(51,76,71,96,67,92,63,88,59,84,55,80)(52,93,56,73,60,77,64,81,68,85,72,89), (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), (1,88,19,82,13,76,7,94)(2,66,20,60,14,54,8,72)(3,74,21,92,15,86,9,80)(4,52,22,70,16,64,10,58)(5,84,23,78,17,96,11,90)(6,62,24,56,18,50,12,68)(25,81,43,75,37,93,31,87)(26,59,44,53,38,71,32,65)(27,91,45,85,39,79,33,73)(28,69,46,63,40,57,34,51)(29,77,47,95,41,89,35,83)(30,55,48,49,42,67,36,61) );
G=PermutationGroup([[(1,26,5,30,9,34,13,38,17,42,21,46),(2,35,22,31,18,27,14,47,10,43,6,39),(3,28,7,32,11,36,15,40,19,44,23,48),(4,37,24,33,20,29,16,25,12,45,8,41),(49,74,69,94,65,90,61,86,57,82,53,78),(50,91,54,95,58,75,62,79,66,83,70,87),(51,76,71,96,67,92,63,88,59,84,55,80),(52,93,56,73,60,77,64,81,68,85,72,89)], [(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)], [(1,88,19,82,13,76,7,94),(2,66,20,60,14,54,8,72),(3,74,21,92,15,86,9,80),(4,52,22,70,16,64,10,58),(5,84,23,78,17,96,11,90),(6,62,24,56,18,50,12,68),(25,81,43,75,37,93,31,87),(26,59,44,53,38,71,32,65),(27,91,45,85,39,79,33,73),(28,69,46,63,40,57,34,51),(29,77,47,95,41,89,35,83),(30,55,48,49,42,67,36,61)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 8A | ··· | 8H | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | - | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C8 | S3 | D4 | Q8 | D6 | M4(2) | Dic6 | C3⋊D4 | C4×S3 | S3×C8 | C8⋊S3 | S32 | D6⋊S3 | C32⋊2Q8 | C6.D6 | C12.29D6 | C12.31D6 |
| kernel | C12.15Dic6 | C6×C3⋊C8 | C4×C3⋊Dic3 | C2×C3⋊Dic3 | C3⋊Dic3 | C2×C3⋊C8 | C3×C12 | C3×C12 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C6 | C6 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C12.15Dic6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 22 | 0 | 0 |
| 0 | 0 | 51 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 23 | 28 | 0 | 0 | 0 | 0 |
| 28 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 51 | 0 | 0 |
| 0 | 0 | 51 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,51,0,0,0,0,22,22,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[23,28,0,0,0,0,28,50,0,0,0,0,0,0,0,51,0,0,0,0,51,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C12.15Dic6 in GAP, Magma, Sage, TeX
C_{12}._{15}{\rm Dic}_6 % in TeX
G:=Group("C12.15Dic6"); // GroupNames label
G:=SmallGroup(288,220);
// by ID
G=gap.SmallGroup(288,220);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,36,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^6*b^6,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^3*b^11>;
// generators/relations