metabelian, supersoluble, monomial
Aliases: C12.73D12, C6.7Dic12, C62.22D4, (C3×C6).5Q16, C12.41(C4×S3), C6.5(D6⋊C4), (C2×C6).52D12, (C3×C12).37D4, C4⋊Dic3.1S3, C32⋊4Q8⋊6C4, (C2×C12).275D6, C6.2(D4.S3), (C3×C6).12SD16, C6.2(C3⋊Q16), C6.11(C24⋊C2), C3⋊1(C6.SD16), C12.33(C3⋊D4), (C6×C12).29C22, C3⋊1(C2.Dic12), C4.2(C6.D6), C32⋊6(Q8⋊C4), C4.15(C3⋊D12), C2.2(C32⋊3Q16), C2.2(D12.S3), C2.6(C6.D12), C22.15(C3⋊D12), (C6×C3⋊C8).2C2, (C2×C4).55S32, (C2×C3⋊C8).2S3, (C3×C12).27(C2×C4), (C3×C4⋊Dic3).2C2, (C2×C6).33(C3⋊D4), (C2×C32⋊4Q8).8C2, (C3×C6).34(C22⋊C4), SmallGroup(288,215)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.73D12
G = < a,b,c | a12=b12=1, c2=a6, bab-1=cac-1=a-1, cbc-1=a9b-1 >
Subgroups: 386 in 103 conjugacy classes, 40 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, C2×Dic3, C2×C12, C2×C12, Q8⋊C4, C3×Dic3, C3⋊Dic3, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6, C3×C3⋊C8, C6×Dic3, C32⋊4Q8, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C6.SD16, C2.Dic12, C6×C3⋊C8, C3×C4⋊Dic3, C2×C32⋊4Q8, C12.73D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, SD16, Q16, C4×S3, D12, C3⋊D4, Q8⋊C4, S32, C24⋊C2, Dic12, D6⋊C4, D4.S3, C3⋊Q16, C6.D6, C3⋊D12, C6.SD16, C2.Dic12, D12.S3, C32⋊3Q16, C6.D12, C12.73D12
►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 52 72 27 5 60 64 35 9 56 68 31)(2 51 61 26 6 59 65 34 10 55 69 30)(3 50 62 25 7 58 66 33 11 54 70 29)(4 49 63 36 8 57 67 32 12 53 71 28)(13 75 38 86 17 83 42 94 21 79 46 90)(14 74 39 85 18 82 43 93 22 78 47 89)(15 73 40 96 19 81 44 92 23 77 48 88)(16 84 41 95 20 80 45 91 24 76 37 87)
(1 41 7 47)(2 40 8 46)(3 39 9 45)(4 38 10 44)(5 37 11 43)(6 48 12 42)(13 69 19 63)(14 68 20 62)(15 67 21 61)(16 66 22 72)(17 65 23 71)(18 64 24 70)(25 92 31 86)(26 91 32 85)(27 90 33 96)(28 89 34 95)(29 88 35 94)(30 87 36 93)(49 78 55 84)(50 77 56 83)(51 76 57 82)(52 75 58 81)(53 74 59 80)(54 73 60 79)
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), (1,52,72,27,5,60,64,35,9,56,68,31)(2,51,61,26,6,59,65,34,10,55,69,30)(3,50,62,25,7,58,66,33,11,54,70,29)(4,49,63,36,8,57,67,32,12,53,71,28)(13,75,38,86,17,83,42,94,21,79,46,90)(14,74,39,85,18,82,43,93,22,78,47,89)(15,73,40,96,19,81,44,92,23,77,48,88)(16,84,41,95,20,80,45,91,24,76,37,87), (1,41,7,47)(2,40,8,46)(3,39,9,45)(4,38,10,44)(5,37,11,43)(6,48,12,42)(13,69,19,63)(14,68,20,62)(15,67,21,61)(16,66,22,72)(17,65,23,71)(18,64,24,70)(25,92,31,86)(26,91,32,85)(27,90,33,96)(28,89,34,95)(29,88,35,94)(30,87,36,93)(49,78,55,84)(50,77,56,83)(51,76,57,82)(52,75,58,81)(53,74,59,80)(54,73,60,79)>;
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), (1,52,72,27,5,60,64,35,9,56,68,31)(2,51,61,26,6,59,65,34,10,55,69,30)(3,50,62,25,7,58,66,33,11,54,70,29)(4,49,63,36,8,57,67,32,12,53,71,28)(13,75,38,86,17,83,42,94,21,79,46,90)(14,74,39,85,18,82,43,93,22,78,47,89)(15,73,40,96,19,81,44,92,23,77,48,88)(16,84,41,95,20,80,45,91,24,76,37,87), (1,41,7,47)(2,40,8,46)(3,39,9,45)(4,38,10,44)(5,37,11,43)(6,48,12,42)(13,69,19,63)(14,68,20,62)(15,67,21,61)(16,66,22,72)(17,65,23,71)(18,64,24,70)(25,92,31,86)(26,91,32,85)(27,90,33,96)(28,89,34,95)(29,88,35,94)(30,87,36,93)(49,78,55,84)(50,77,56,83)(51,76,57,82)(52,75,58,81)(53,74,59,80)(54,73,60,79) );
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)], [(1,52,72,27,5,60,64,35,9,56,68,31),(2,51,61,26,6,59,65,34,10,55,69,30),(3,50,62,25,7,58,66,33,11,54,70,29),(4,49,63,36,8,57,67,32,12,53,71,28),(13,75,38,86,17,83,42,94,21,79,46,90),(14,74,39,85,18,82,43,93,22,78,47,89),(15,73,40,96,19,81,44,92,23,77,48,88),(16,84,41,95,20,80,45,91,24,76,37,87)], [(1,41,7,47),(2,40,8,46),(3,39,9,45),(4,38,10,44),(5,37,11,43),(6,48,12,42),(13,69,19,63),(14,68,20,62),(15,67,21,61),(16,66,22,72),(17,65,23,71),(18,64,24,70),(25,92,31,86),(26,91,32,85),(27,90,33,96),(28,89,34,95),(29,88,35,94),(30,87,36,93),(49,78,55,84),(50,77,56,83),(51,76,57,82),(52,75,58,81),(53,74,59,80),(54,73,60,79)]])
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 | D6 | SD16 | Q16 | C4×S3 | D12 | C3⋊D4 | D12 | C3⋊D4 | C24⋊C2 | Dic12 | S32 | D4.S3 | C3⋊Q16 | C6.D6 | C3⋊D12 | C3⋊D12 | D12.S3 | C32⋊3Q16 |
| kernel | C12.73D12 | C6×C3⋊C8 | C3×C4⋊Dic3 | C2×C32⋊4Q8 | C32⋊4Q8 | C2×C3⋊C8 | C4⋊Dic3 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C12 | 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 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C12.73D12 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 14 | 66 |
| 0 | 0 | 7 | 7 |
| 0 | 27 | 0 | 0 |
| 46 | 46 | 0 | 0 |
| 0 | 0 | 50 | 5 |
| 0 | 0 | 55 | 23 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 30 | 13 |
| 0 | 0 | 43 | 43 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,14,7,0,0,66,7],[0,46,0,0,27,46,0,0,0,0,50,55,0,0,5,23],[0,72,0,0,72,0,0,0,0,0,30,43,0,0,13,43] >;
C12.73D12 in GAP, Magma, Sage, TeX
C_{12}._{73}D_{12} % in TeX
G:=Group("C12.73D12"); // GroupNames label
G:=SmallGroup(288,215);
// by ID
G=gap.SmallGroup(288,215);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,92,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=1,c^2=a^6,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^-1>;
// generators/relations