metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.8D4, C23.2Dic15, (C6×D4).2D5, (C2×C4).3D30, (D4×C10).2S3, (C2×D4).2D15, (D4×C30).2C2, (C2×C20).73D6, (C2×C12).74D10, C3⋊2(C20.D4), C5⋊3(C12.D4), (C22×C30).2C4, C60.7C4⋊15C2, C20.20(C3⋊D4), C15⋊10(C4.D4), C4.13(C15⋊7D4), C12.22(C5⋊D4), (C2×C60).59C22, (C22×C6).2Dic5, C6.15(C23.D5), C22.2(C2×Dic15), (C22×C10).5Dic3, C30.103(C22⋊C4), C2.4(C30.38D4), C10.26(C6.D4), (C2×C30).173(C2×C4), (C2×C6).30(C2×Dic5), (C2×C10).50(C2×Dic3), SmallGroup(480,193)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.8D4
G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a-1, cac-1=a29, cbc-1=a15b3 >
Subgroups: 308 in 92 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, C23, C10, C10, C12, C2×C6, C2×C6, C15, M4(2), C2×D4, C20, C2×C10, C2×C10, C3⋊C8, C2×C12, C3×D4, C22×C6, C30, C30, C4.D4, C5⋊2C8, C2×C20, C5×D4, C22×C10, C4.Dic3, C6×D4, C60, C2×C30, C2×C30, C4.Dic5, D4×C10, C12.D4, C15⋊3C8, C2×C60, D4×C15, C22×C30, C20.D4, C60.7C4, D4×C30, C60.8D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, Dic5, D10, C2×Dic3, C3⋊D4, D15, C4.D4, C2×Dic5, C5⋊D4, C6.D4, Dic15, D30, C23.D5, C12.D4, C2×Dic15, C15⋊7D4, C20.D4, C30.38D4, C60.8D4
►On 120 points
Generators in S120
(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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 115 46 70 31 85 16 100)(2 114 47 69 32 84 17 99)(3 113 48 68 33 83 18 98)(4 112 49 67 34 82 19 97)(5 111 50 66 35 81 20 96)(6 110 51 65 36 80 21 95)(7 109 52 64 37 79 22 94)(8 108 53 63 38 78 23 93)(9 107 54 62 39 77 24 92)(10 106 55 61 40 76 25 91)(11 105 56 120 41 75 26 90)(12 104 57 119 42 74 27 89)(13 103 58 118 43 73 28 88)(14 102 59 117 44 72 29 87)(15 101 60 116 45 71 30 86)
(1 85 46 70 31 115 16 100)(2 114 47 99 32 84 17 69)(3 83 48 68 33 113 18 98)(4 112 49 97 34 82 19 67)(5 81 50 66 35 111 20 96)(6 110 51 95 36 80 21 65)(7 79 52 64 37 109 22 94)(8 108 53 93 38 78 23 63)(9 77 54 62 39 107 24 92)(10 106 55 91 40 76 25 61)(11 75 56 120 41 105 26 90)(12 104 57 89 42 74 27 119)(13 73 58 118 43 103 28 88)(14 102 59 87 44 72 29 117)(15 71 60 116 45 101 30 86)
G:=sub<Sym(120)| (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,115,46,70,31,85,16,100)(2,114,47,69,32,84,17,99)(3,113,48,68,33,83,18,98)(4,112,49,67,34,82,19,97)(5,111,50,66,35,81,20,96)(6,110,51,65,36,80,21,95)(7,109,52,64,37,79,22,94)(8,108,53,63,38,78,23,93)(9,107,54,62,39,77,24,92)(10,106,55,61,40,76,25,91)(11,105,56,120,41,75,26,90)(12,104,57,119,42,74,27,89)(13,103,58,118,43,73,28,88)(14,102,59,117,44,72,29,87)(15,101,60,116,45,71,30,86), (1,85,46,70,31,115,16,100)(2,114,47,99,32,84,17,69)(3,83,48,68,33,113,18,98)(4,112,49,97,34,82,19,67)(5,81,50,66,35,111,20,96)(6,110,51,95,36,80,21,65)(7,79,52,64,37,109,22,94)(8,108,53,93,38,78,23,63)(9,77,54,62,39,107,24,92)(10,106,55,91,40,76,25,61)(11,75,56,120,41,105,26,90)(12,104,57,89,42,74,27,119)(13,73,58,118,43,103,28,88)(14,102,59,87,44,72,29,117)(15,71,60,116,45,101,30,86)>;
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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,115,46,70,31,85,16,100)(2,114,47,69,32,84,17,99)(3,113,48,68,33,83,18,98)(4,112,49,67,34,82,19,97)(5,111,50,66,35,81,20,96)(6,110,51,65,36,80,21,95)(7,109,52,64,37,79,22,94)(8,108,53,63,38,78,23,93)(9,107,54,62,39,77,24,92)(10,106,55,61,40,76,25,91)(11,105,56,120,41,75,26,90)(12,104,57,119,42,74,27,89)(13,103,58,118,43,73,28,88)(14,102,59,117,44,72,29,87)(15,101,60,116,45,71,30,86), (1,85,46,70,31,115,16,100)(2,114,47,99,32,84,17,69)(3,83,48,68,33,113,18,98)(4,112,49,97,34,82,19,67)(5,81,50,66,35,111,20,96)(6,110,51,95,36,80,21,65)(7,79,52,64,37,109,22,94)(8,108,53,93,38,78,23,63)(9,77,54,62,39,107,24,92)(10,106,55,91,40,76,25,61)(11,75,56,120,41,105,26,90)(12,104,57,89,42,74,27,119)(13,73,58,118,43,103,28,88)(14,102,59,87,44,72,29,117)(15,71,60,116,45,101,30,86) );
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),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,115,46,70,31,85,16,100),(2,114,47,69,32,84,17,99),(3,113,48,68,33,83,18,98),(4,112,49,67,34,82,19,97),(5,111,50,66,35,81,20,96),(6,110,51,65,36,80,21,95),(7,109,52,64,37,79,22,94),(8,108,53,63,38,78,23,93),(9,107,54,62,39,77,24,92),(10,106,55,61,40,76,25,91),(11,105,56,120,41,75,26,90),(12,104,57,119,42,74,27,89),(13,103,58,118,43,73,28,88),(14,102,59,117,44,72,29,87),(15,101,60,116,45,71,30,86)], [(1,85,46,70,31,115,16,100),(2,114,47,99,32,84,17,69),(3,83,48,68,33,113,18,98),(4,112,49,97,34,82,19,67),(5,81,50,66,35,111,20,96),(6,110,51,95,36,80,21,65),(7,79,52,64,37,109,22,94),(8,108,53,93,38,78,23,63),(9,77,54,62,39,107,24,92),(10,106,55,91,40,76,25,61),(11,75,56,120,41,105,26,90),(12,104,57,89,42,74,27,119),(13,73,58,118,43,103,28,88),(14,102,59,87,44,72,29,117),(15,71,60,116,45,101,30,86)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 60 | 60 | 60 | 60 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | Dic3 | D10 | Dic5 | C3⋊D4 | D15 | C5⋊D4 | D30 | Dic15 | C15⋊7D4 | C4.D4 | C12.D4 | C20.D4 | C60.8D4 |
| kernel | C60.8D4 | C60.7C4 | D4×C30 | C22×C30 | D4×C10 | C60 | C6×D4 | C2×C20 | C22×C10 | C2×C12 | C22×C6 | C20 | C2×D4 | C12 | C2×C4 | C23 | C4 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C60.8D4 ►in GL4(𝔽241) generated by
| 0 | 24 | 0 | 0 |
| 217 | 0 | 0 | 0 |
| 0 | 0 | 0 | 231 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
G:=sub<GL(4,GF(241))| [0,217,0,0,24,0,0,0,0,0,0,10,0,0,231,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,240,0,0],[0,0,0,240,0,0,1,0,1,0,0,0,0,1,0,0] >;
C60.8D4 in GAP, Magma, Sage, TeX
C_{60}._8D_4 % in TeX
G:=Group("C60.8D4"); // GroupNames label
G:=SmallGroup(480,193);
// by ID
G=gap.SmallGroup(480,193);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,100,675,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^45,b*a*b^-1=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^15*b^3>;
// generators/relations