metabelian, supersoluble, monomial
Aliases: C12.77D12, D6⋊(C3⋊C8), (S3×C6)⋊1C8, C3⋊3(D6⋊C8), C6.20(S3×C8), C6.31(D6⋊C4), (C2×C12).295D6, (C3×C12).107D4, C32⋊4(C22⋊C8), C62.25(C2×C4), (C6×Dic3).4C4, C6.12(C8⋊S3), (C3×C6).5M4(2), C12.94(C3⋊D4), C2.1(D6⋊Dic3), C6.2(C4.Dic3), C4.16(D6⋊S3), C4.26(C3⋊D12), C3⋊1(C12.55D4), (C6×C12).200C22, (C2×Dic3).3Dic3, C6.1(C6.D4), C2.2(D6.Dic3), (C22×S3).2Dic3, C22.10(S3×Dic3), (C6×C3⋊C8)⋊1C2, (C2×C3⋊C8)⋊9S3, C6.4(C2×C3⋊C8), C2.4(S3×C3⋊C8), (S3×C2×C4).7S3, (S3×C2×C6).3C4, (C2×C4).128S32, (S3×C2×C12).16C2, (C2×C6).65(C4×S3), (C3×C6).18(C2×C8), (C2×C32⋊4C8)⋊14C2, (C2×C6).14(C2×Dic3), (C3×C6).23(C22⋊C4), SmallGroup(288,204)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.77D12
G = < a,b,c | a12=1, b12=a6, c2=a9, bab-1=cac-1=a5, cbc-1=a3b11 >
Subgroups: 306 in 111 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C22×C4, C3×S3, C3×C6, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C2×C3⋊C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C22×C12, C3×C3⋊C8, C32⋊4C8, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, D6⋊C8, C12.55D4, C6×C3⋊C8, C2×C32⋊4C8, S3×C2×C12, C12.77D12
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Dic3, D6, C22⋊C4, C2×C8, M4(2), C3⋊C8, C4×S3, D12, C2×Dic3, C3⋊D4, C22⋊C8, S32, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, D6⋊C8, C12.55D4, S3×C3⋊C8, D6.Dic3, D6⋊Dic3, C12.77D12
►On 96 points
(1 74 21 94 17 90 13 86 9 82 5 78)(2 91 6 95 10 75 14 79 18 83 22 87)(3 76 23 96 19 92 15 88 11 84 7 80)(4 93 8 73 12 77 16 81 20 85 24 89)(25 67 45 63 41 59 37 55 33 51 29 71)(26 60 30 64 34 68 38 72 42 52 46 56)(27 69 47 65 43 61 39 57 35 53 31 49)(28 62 32 66 36 70 40 50 44 54 48 58)
(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 51 82 37 13 63 94 25)(2 36 83 62 14 48 95 50)(3 61 84 47 15 49 96 35)(4 46 85 72 16 34 73 60)(5 71 86 33 17 59 74 45)(6 32 87 58 18 44 75 70)(7 57 88 43 19 69 76 31)(8 42 89 68 20 30 77 56)(9 67 90 29 21 55 78 41)(10 28 91 54 22 40 79 66)(11 53 92 39 23 65 80 27)(12 38 93 64 24 26 81 52)
G:=sub<Sym(96)| (1,74,21,94,17,90,13,86,9,82,5,78)(2,91,6,95,10,75,14,79,18,83,22,87)(3,76,23,96,19,92,15,88,11,84,7,80)(4,93,8,73,12,77,16,81,20,85,24,89)(25,67,45,63,41,59,37,55,33,51,29,71)(26,60,30,64,34,68,38,72,42,52,46,56)(27,69,47,65,43,61,39,57,35,53,31,49)(28,62,32,66,36,70,40,50,44,54,48,58), (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), (1,51,82,37,13,63,94,25)(2,36,83,62,14,48,95,50)(3,61,84,47,15,49,96,35)(4,46,85,72,16,34,73,60)(5,71,86,33,17,59,74,45)(6,32,87,58,18,44,75,70)(7,57,88,43,19,69,76,31)(8,42,89,68,20,30,77,56)(9,67,90,29,21,55,78,41)(10,28,91,54,22,40,79,66)(11,53,92,39,23,65,80,27)(12,38,93,64,24,26,81,52)>;
G:=Group( (1,74,21,94,17,90,13,86,9,82,5,78)(2,91,6,95,10,75,14,79,18,83,22,87)(3,76,23,96,19,92,15,88,11,84,7,80)(4,93,8,73,12,77,16,81,20,85,24,89)(25,67,45,63,41,59,37,55,33,51,29,71)(26,60,30,64,34,68,38,72,42,52,46,56)(27,69,47,65,43,61,39,57,35,53,31,49)(28,62,32,66,36,70,40,50,44,54,48,58), (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), (1,51,82,37,13,63,94,25)(2,36,83,62,14,48,95,50)(3,61,84,47,15,49,96,35)(4,46,85,72,16,34,73,60)(5,71,86,33,17,59,74,45)(6,32,87,58,18,44,75,70)(7,57,88,43,19,69,76,31)(8,42,89,68,20,30,77,56)(9,67,90,29,21,55,78,41)(10,28,91,54,22,40,79,66)(11,53,92,39,23,65,80,27)(12,38,93,64,24,26,81,52) );
G=PermutationGroup([[(1,74,21,94,17,90,13,86,9,82,5,78),(2,91,6,95,10,75,14,79,18,83,22,87),(3,76,23,96,19,92,15,88,11,84,7,80),(4,93,8,73,12,77,16,81,20,85,24,89),(25,67,45,63,41,59,37,55,33,51,29,71),(26,60,30,64,34,68,38,72,42,52,46,56),(27,69,47,65,43,61,39,57,35,53,31,49),(28,62,32,66,36,70,40,50,44,54,48,58)], [(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)], [(1,51,82,37,13,63,94,25),(2,36,83,62,14,48,95,50),(3,61,84,47,15,49,96,35),(4,46,85,72,16,34,73,60),(5,71,86,33,17,59,74,45),(6,32,87,58,18,44,75,70),(7,57,88,43,19,69,76,31),(8,42,89,68,20,30,77,56),(9,67,90,29,21,55,78,41),(10,28,91,54,22,40,79,66),(11,53,92,39,23,65,80,27),(12,38,93,64,24,26,81,52)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | 12N | 12O | 12P | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 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 |
| type | + | + | + | + | + | + | + | - | + | - | + | + | - | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | S3 | D4 | Dic3 | D6 | Dic3 | M4(2) | D12 | C3⋊D4 | C3⋊C8 | C4×S3 | S3×C8 | C8⋊S3 | C4.Dic3 | S32 | D6⋊S3 | C3⋊D12 | S3×Dic3 | S3×C3⋊C8 | D6.Dic3 |
| kernel | C12.77D12 | C6×C3⋊C8 | C2×C32⋊4C8 | S3×C2×C12 | C6×Dic3 | S3×C2×C6 | S3×C6 | C2×C3⋊C8 | S3×C2×C4 | C3×C12 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | C12 | C12 | D6 | C2×C6 | C6 | C6 | C6 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 6 | 4 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C12.77D12 ►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 | 51 | 22 | 0 | 0 |
| 0 | 0 | 51 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 30 | 14 | 0 | 0 | 0 | 0 |
| 14 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 51 | 0 | 0 |
| 0 | 0 | 51 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
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],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,51,51,0,0,0,0,22,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[30,14,0,0,0,0,14,43,0,0,0,0,0,0,0,51,0,0,0,0,51,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C12.77D12 in GAP, Magma, Sage, TeX
C_{12}._{77}D_{12} % in TeX
G:=Group("C12.77D12"); // GroupNames label
G:=SmallGroup(288,204);
// by ID
G=gap.SmallGroup(288,204);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;
// generators/relations