metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.58D8, C24.41D4, D8.1Dic3, Q16.1Dic3, C4○D8.1S3, (C3×D8).1C4, C24.15(C2×C4), (C2×C8).250D6, (C3×Q16).1C4, C8.Dic3⋊5C2, C8.9(C2×Dic3), C4.31(D4⋊S3), (C2×C12).119D4, C3⋊3(D8.C4), C8.31(C3⋊D4), (C2×C6).12SD16, (C2×C24).38C22, C6.31(D4⋊C4), C12.18(C22⋊C4), C4.6(C6.D4), C22.1(D4.S3), C2.11(D4⋊Dic3), (C2×C3⋊C16)⋊2C2, (C3×C4○D8).1C2, (C2×C4).121(C3⋊D4), SmallGroup(192,126)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.58D8
G = < a,b,c | a12=1, b8=a6, c2=a9, bab-1=cac-1=a5, cbc-1=a3b7 >
Smallest permutation representation of C12.58D8
►On 96 points
Generators in S96
(1 72 88 22 51 37 9 80 96 30 59 45)(2 38 60 23 81 73 10 46 52 31 89 65)(3 74 90 24 53 39 11 66 82 32 61 47)(4 40 62 25 83 75 12 48 54 17 91 67)(5 76 92 26 55 41 13 68 84 18 63 33)(6 42 64 27 85 77 14 34 56 19 93 69)(7 78 94 28 57 43 15 70 86 20 49 35)(8 44 50 29 87 79 16 36 58 21 95 71)
(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 29 30 8 9 21 22 16)(2 7 31 20 10 15 23 28)(3 19 32 14 11 27 24 6)(4 13 17 26 12 5 25 18)(33 91 84 48 41 83 92 40)(34 47 85 82 42 39 93 90)(35 81 86 38 43 89 94 46)(36 37 87 88 44 45 95 96)(49 65 70 52 57 73 78 60)(50 51 71 72 58 59 79 80)(53 77 74 64 61 69 66 56)(54 63 75 68 62 55 67 76)
G:=sub<Sym(96)| (1,72,88,22,51,37,9,80,96,30,59,45)(2,38,60,23,81,73,10,46,52,31,89,65)(3,74,90,24,53,39,11,66,82,32,61,47)(4,40,62,25,83,75,12,48,54,17,91,67)(5,76,92,26,55,41,13,68,84,18,63,33)(6,42,64,27,85,77,14,34,56,19,93,69)(7,78,94,28,57,43,15,70,86,20,49,35)(8,44,50,29,87,79,16,36,58,21,95,71), (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), (1,29,30,8,9,21,22,16)(2,7,31,20,10,15,23,28)(3,19,32,14,11,27,24,6)(4,13,17,26,12,5,25,18)(33,91,84,48,41,83,92,40)(34,47,85,82,42,39,93,90)(35,81,86,38,43,89,94,46)(36,37,87,88,44,45,95,96)(49,65,70,52,57,73,78,60)(50,51,71,72,58,59,79,80)(53,77,74,64,61,69,66,56)(54,63,75,68,62,55,67,76)>;
G:=Group( (1,72,88,22,51,37,9,80,96,30,59,45)(2,38,60,23,81,73,10,46,52,31,89,65)(3,74,90,24,53,39,11,66,82,32,61,47)(4,40,62,25,83,75,12,48,54,17,91,67)(5,76,92,26,55,41,13,68,84,18,63,33)(6,42,64,27,85,77,14,34,56,19,93,69)(7,78,94,28,57,43,15,70,86,20,49,35)(8,44,50,29,87,79,16,36,58,21,95,71), (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), (1,29,30,8,9,21,22,16)(2,7,31,20,10,15,23,28)(3,19,32,14,11,27,24,6)(4,13,17,26,12,5,25,18)(33,91,84,48,41,83,92,40)(34,47,85,82,42,39,93,90)(35,81,86,38,43,89,94,46)(36,37,87,88,44,45,95,96)(49,65,70,52,57,73,78,60)(50,51,71,72,58,59,79,80)(53,77,74,64,61,69,66,56)(54,63,75,68,62,55,67,76) );
G=PermutationGroup([(1,72,88,22,51,37,9,80,96,30,59,45),(2,38,60,23,81,73,10,46,52,31,89,65),(3,74,90,24,53,39,11,66,82,32,61,47),(4,40,62,25,83,75,12,48,54,17,91,67),(5,76,92,26,55,41,13,68,84,18,63,33),(6,42,64,27,85,77,14,34,56,19,93,69),(7,78,94,28,57,43,15,70,86,20,49,35),(8,44,50,29,87,79,16,36,58,21,95,71)], [(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)], [(1,29,30,8,9,21,22,16),(2,7,31,20,10,15,23,28),(3,19,32,14,11,27,24,6),(4,13,17,26,12,5,25,18),(33,91,84,48,41,83,92,40),(34,47,85,82,42,39,93,90),(35,81,86,38,43,89,94,46),(36,37,87,88,44,45,95,96),(49,65,70,52,57,73,78,60),(50,51,71,72,58,59,79,80),(53,77,74,64,61,69,66,56),(54,63,75,68,62,55,67,76)])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 12D | 12E | 16A | ··· | 16H | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 8 | 2 | 1 | 1 | 2 | 8 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 24 | 24 | 2 | 2 | 4 | 8 | 8 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | Dic3 | Dic3 | D8 | SD16 | C3⋊D4 | C3⋊D4 | D8.C4 | D4⋊S3 | D4.S3 | C12.58D8 |
| kernel | C12.58D8 | C2×C3⋊C16 | C8.Dic3 | C3×C4○D8 | C3×D8 | C3×Q16 | C4○D8 | C24 | C2×C12 | C2×C8 | D8 | Q16 | C12 | C2×C6 | C8 | C2×C4 | C3 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 4 |
Matrix representation of C12.58D8 ►in GL4(𝔽97) generated by
| 22 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 35 | 0 |
| 0 | 0 | 46 | 61 |
| 46 | 67 | 0 | 0 |
| 15 | 16 | 0 | 0 |
| 0 | 0 | 11 | 2 |
| 0 | 0 | 36 | 86 |
| 46 | 67 | 0 | 0 |
| 31 | 51 | 0 | 0 |
| 0 | 0 | 11 | 2 |
| 0 | 0 | 37 | 86 |
G:=sub<GL(4,GF(97))| [22,0,0,0,0,22,0,0,0,0,35,46,0,0,0,61],[46,15,0,0,67,16,0,0,0,0,11,36,0,0,2,86],[46,31,0,0,67,51,0,0,0,0,11,37,0,0,2,86] >;
C12.58D8 in GAP, Magma, Sage, TeX
C_{12}._{58}D_8 % in TeX
G:=Group("C12.58D8"); // GroupNames label
G:=SmallGroup(192,126);
// by ID
G=gap.SmallGroup(192,126);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,184,675,346,192,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^8=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^3*b^7>;
// generators/relations