metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.28D4, C12.5D20, (C6×D20).1C2, (C2×D20).6S3, (C2×C20).40D6, C4.Dic3⋊2D5, C60.7C4⋊5C2, (C2×C12).41D10, C15⋊2(C4.D4), C5⋊2(C12.D4), C12.5(C5⋊D4), C4.19(C15⋊D4), C3⋊3(C20.46D4), C4.19(C3⋊D20), C20.77(C3⋊D4), (C2×C60).29C22, C22.3(D5×Dic3), C30.41(C22⋊C4), C6.24(D10⋊C4), (C22×D5).1Dic3, C2.3(D10⋊Dic3), C10.13(C6.D4), (D5×C2×C6).1C4, (C2×C4).1(S3×D5), (C2×C6).45(C4×D5), (C2×C30).80(C2×C4), (C5×C4.Dic3)⋊1C2, (C2×C10).21(C2×Dic3), SmallGroup(480,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.28D4
G = < a,b,c | a12=1, b20=a6, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a3b19 >
Subgroups: 476 in 92 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, M4(2), C2×D4, C20, D10, C2×C10, C3⋊C8, C2×C12, C3×D4, C22×C6, C3×D5, C30, C30, C4.D4, C5⋊2C8, C40, D20, C2×C20, C22×D5, C4.Dic3, C4.Dic3, C6×D4, C60, C6×D5, C2×C30, C4.Dic5, C5×M4(2), C2×D20, C12.D4, C5×C3⋊C8, C15⋊3C8, C3×D20, C2×C60, D5×C2×C6, C20.46D4, C5×C4.Dic3, C60.7C4, C6×D20, C60.28D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, D10, C2×Dic3, C3⋊D4, C4.D4, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, C12.D4, D5×Dic3, C15⋊D4, C3⋊D20, C20.46D4, D10⋊Dic3, C60.28D4
►On 120 points
(1 93 49 31 83 79 21 113 69 11 103 59)(2 60 104 12 70 114 22 80 84 32 50 94)(3 95 51 33 85 41 23 115 71 13 105 61)(4 62 106 14 72 116 24 42 86 34 52 96)(5 97 53 35 87 43 25 117 73 15 107 63)(6 64 108 16 74 118 26 44 88 36 54 98)(7 99 55 37 89 45 27 119 75 17 109 65)(8 66 110 18 76 120 28 46 90 38 56 100)(9 101 57 39 91 47 29 81 77 19 111 67)(10 68 112 20 78 82 30 48 92 40 58 102)
(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 30 11 20 21 10 31 40)(2 39 32 9 22 19 12 29)(3 28 13 18 23 8 33 38)(4 37 34 7 24 17 14 27)(5 26 15 16 25 6 35 36)(41 46 51 76 61 66 71 56)(42 55 72 65 62 75 52 45)(43 44 53 74 63 64 73 54)(47 80 57 70 67 60 77 50)(48 49 78 59 68 69 58 79)(81 114 91 104 101 94 111 84)(82 83 112 93 102 103 92 113)(85 110 95 100 105 90 115 120)(86 119 116 89 106 99 96 109)(87 108 97 98 107 88 117 118)
G:=sub<Sym(120)| (1,93,49,31,83,79,21,113,69,11,103,59)(2,60,104,12,70,114,22,80,84,32,50,94)(3,95,51,33,85,41,23,115,71,13,105,61)(4,62,106,14,72,116,24,42,86,34,52,96)(5,97,53,35,87,43,25,117,73,15,107,63)(6,64,108,16,74,118,26,44,88,36,54,98)(7,99,55,37,89,45,27,119,75,17,109,65)(8,66,110,18,76,120,28,46,90,38,56,100)(9,101,57,39,91,47,29,81,77,19,111,67)(10,68,112,20,78,82,30,48,92,40,58,102), (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), (1,30,11,20,21,10,31,40)(2,39,32,9,22,19,12,29)(3,28,13,18,23,8,33,38)(4,37,34,7,24,17,14,27)(5,26,15,16,25,6,35,36)(41,46,51,76,61,66,71,56)(42,55,72,65,62,75,52,45)(43,44,53,74,63,64,73,54)(47,80,57,70,67,60,77,50)(48,49,78,59,68,69,58,79)(81,114,91,104,101,94,111,84)(82,83,112,93,102,103,92,113)(85,110,95,100,105,90,115,120)(86,119,116,89,106,99,96,109)(87,108,97,98,107,88,117,118)>;
G:=Group( (1,93,49,31,83,79,21,113,69,11,103,59)(2,60,104,12,70,114,22,80,84,32,50,94)(3,95,51,33,85,41,23,115,71,13,105,61)(4,62,106,14,72,116,24,42,86,34,52,96)(5,97,53,35,87,43,25,117,73,15,107,63)(6,64,108,16,74,118,26,44,88,36,54,98)(7,99,55,37,89,45,27,119,75,17,109,65)(8,66,110,18,76,120,28,46,90,38,56,100)(9,101,57,39,91,47,29,81,77,19,111,67)(10,68,112,20,78,82,30,48,92,40,58,102), (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), (1,30,11,20,21,10,31,40)(2,39,32,9,22,19,12,29)(3,28,13,18,23,8,33,38)(4,37,34,7,24,17,14,27)(5,26,15,16,25,6,35,36)(41,46,51,76,61,66,71,56)(42,55,72,65,62,75,52,45)(43,44,53,74,63,64,73,54)(47,80,57,70,67,60,77,50)(48,49,78,59,68,69,58,79)(81,114,91,104,101,94,111,84)(82,83,112,93,102,103,92,113)(85,110,95,100,105,90,115,120)(86,119,116,89,106,99,96,109)(87,108,97,98,107,88,117,118) );
G=PermutationGroup([[(1,93,49,31,83,79,21,113,69,11,103,59),(2,60,104,12,70,114,22,80,84,32,50,94),(3,95,51,33,85,41,23,115,71,13,105,61),(4,62,106,14,72,116,24,42,86,34,52,96),(5,97,53,35,87,43,25,117,73,15,107,63),(6,64,108,16,74,118,26,44,88,36,54,98),(7,99,55,37,89,45,27,119,75,17,109,65),(8,66,110,18,76,120,28,46,90,38,56,100),(9,101,57,39,91,47,29,81,77,19,111,67),(10,68,112,20,78,82,30,48,92,40,58,102)], [(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)], [(1,30,11,20,21,10,31,40),(2,39,32,9,22,19,12,29),(3,28,13,18,23,8,33,38),(4,37,34,7,24,17,14,27),(5,26,15,16,25,6,35,36),(41,46,51,76,61,66,71,56),(42,55,72,65,62,75,52,45),(43,44,53,74,63,64,73,54),(47,80,57,70,67,60,77,50),(48,49,78,59,68,69,58,79),(81,114,91,104,101,94,111,84),(82,83,112,93,102,103,92,113),(85,110,95,100,105,90,115,120),(86,119,116,89,106,99,96,109),(87,108,97,98,107,88,117,118)]])
57 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 | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | ··· | 30F | 40A | ··· | 40H | 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 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 12 | 12 | 60 | 60 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 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 | D4 | D5 | D6 | Dic3 | D10 | C3⋊D4 | D20 | C5⋊D4 | C4×D5 | C4.D4 | S3×D5 | C12.D4 | C15⋊D4 | C3⋊D20 | D5×Dic3 | C20.46D4 | C60.28D4 |
| kernel | C60.28D4 | C5×C4.Dic3 | C60.7C4 | C6×D20 | D5×C2×C6 | C2×D20 | C60 | C4.Dic3 | C2×C20 | C22×D5 | C2×C12 | C20 | C12 | C12 | C2×C6 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C60.28D4 ►in GL4(𝔽241) generated by
| 178 | 45 | 0 | 0 |
| 196 | 63 | 0 | 0 |
| 0 | 0 | 222 | 48 |
| 0 | 0 | 193 | 19 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 51 |
| 238 | 197 | 0 | 0 |
| 44 | 163 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 238 | 197 | 0 | 0 |
| 44 | 3 | 0 | 0 |
G:=sub<GL(4,GF(241))| [178,196,0,0,45,63,0,0,0,0,222,193,0,0,48,19],[0,0,238,44,0,0,197,163,0,240,0,0,1,51,0,0],[0,0,238,44,0,0,197,3,0,1,0,0,1,0,0,0] >;
C60.28D4 in GAP, Magma, Sage, TeX
C_{60}._{28}D_4 % in TeX
G:=Group("C60.28D4"); // GroupNames label
G:=SmallGroup(480,34);
// by ID
G=gap.SmallGroup(480,34);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,100,346,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^20=a^6,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^19>;
// generators/relations