metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.53D4, C20.5D12, (C2×D12).5D5, (C2×C20).41D6, C4.Dic5⋊2S3, C60.7C4⋊2C2, (C10×D12).1C2, (C2×C12).42D10, C20.5(C3⋊D4), C15⋊3(C4.D4), C12.6(C5⋊D4), C3⋊1(C20.D4), C10.39(D6⋊C4), (C22×S3).Dic5, C4.19(C5⋊D12), C5⋊4(M4(2)⋊S3), C4.12(C15⋊D4), C2.3(D6⋊Dic5), (C2×C60).22C22, C6.2(C23.D5), C22.3(S3×Dic5), C30.42(C22⋊C4), (S3×C2×C10).1C4, (C2×C4).2(S3×D5), (C2×C10).70(C4×S3), (C2×C30).81(C2×C4), (C3×C4.Dic5)⋊1C2, (C2×C6).1(C2×Dic5), SmallGroup(480,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.53D4
G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a-1, cac-1=a49, cbc-1=a15b3 >
Subgroups: 380 in 92 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6, C8 [×2], C2×C4, D4 [×2], C23 [×2], C10, C10 [×3], C12 [×2], D6 [×4], C2×C6, C15, M4(2) [×2], C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8, C24, D12 [×2], C2×C12, C22×S3 [×2], C5×S3 [×2], C30, C30, C4.D4, C5⋊2C8 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C4.Dic3, C3×M4(2), C2×D12, C60 [×2], S3×C10 [×4], C2×C30, C4.Dic5, C4.Dic5, D4×C10, M4(2)⋊S3, C3×C5⋊2C8, C15⋊3C8, C5×D12 [×2], C2×C60, S3×C2×C10 [×2], C20.D4, C3×C4.Dic5, C60.7C4, C10×D12, C60.53D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, C4.D4, C2×Dic5, C5⋊D4 [×2], D6⋊C4, S3×D5, C23.D5, M4(2)⋊S3, S3×Dic5, C15⋊D4, C5⋊D12, C20.D4, D6⋊Dic5, C60.53D4
►On 120 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 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 95 46 110 31 65 16 80)(2 94 47 109 32 64 17 79)(3 93 48 108 33 63 18 78)(4 92 49 107 34 62 19 77)(5 91 50 106 35 61 20 76)(6 90 51 105 36 120 21 75)(7 89 52 104 37 119 22 74)(8 88 53 103 38 118 23 73)(9 87 54 102 39 117 24 72)(10 86 55 101 40 116 25 71)(11 85 56 100 41 115 26 70)(12 84 57 99 42 114 27 69)(13 83 58 98 43 113 28 68)(14 82 59 97 44 112 29 67)(15 81 60 96 45 111 30 66)
(1 65 46 110 31 95 16 80)(2 114 47 99 32 84 17 69)(3 103 48 88 33 73 18 118)(4 92 49 77 34 62 19 107)(5 81 50 66 35 111 20 96)(6 70 51 115 36 100 21 85)(7 119 52 104 37 89 22 74)(8 108 53 93 38 78 23 63)(9 97 54 82 39 67 24 112)(10 86 55 71 40 116 25 101)(11 75 56 120 41 105 26 90)(12 64 57 109 42 94 27 79)(13 113 58 98 43 83 28 68)(14 102 59 87 44 72 29 117)(15 91 60 76 45 61 30 106)
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), (1,95,46,110,31,65,16,80)(2,94,47,109,32,64,17,79)(3,93,48,108,33,63,18,78)(4,92,49,107,34,62,19,77)(5,91,50,106,35,61,20,76)(6,90,51,105,36,120,21,75)(7,89,52,104,37,119,22,74)(8,88,53,103,38,118,23,73)(9,87,54,102,39,117,24,72)(10,86,55,101,40,116,25,71)(11,85,56,100,41,115,26,70)(12,84,57,99,42,114,27,69)(13,83,58,98,43,113,28,68)(14,82,59,97,44,112,29,67)(15,81,60,96,45,111,30,66), (1,65,46,110,31,95,16,80)(2,114,47,99,32,84,17,69)(3,103,48,88,33,73,18,118)(4,92,49,77,34,62,19,107)(5,81,50,66,35,111,20,96)(6,70,51,115,36,100,21,85)(7,119,52,104,37,89,22,74)(8,108,53,93,38,78,23,63)(9,97,54,82,39,67,24,112)(10,86,55,71,40,116,25,101)(11,75,56,120,41,105,26,90)(12,64,57,109,42,94,27,79)(13,113,58,98,43,83,28,68)(14,102,59,87,44,72,29,117)(15,91,60,76,45,61,30,106)>;
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), (1,95,46,110,31,65,16,80)(2,94,47,109,32,64,17,79)(3,93,48,108,33,63,18,78)(4,92,49,107,34,62,19,77)(5,91,50,106,35,61,20,76)(6,90,51,105,36,120,21,75)(7,89,52,104,37,119,22,74)(8,88,53,103,38,118,23,73)(9,87,54,102,39,117,24,72)(10,86,55,101,40,116,25,71)(11,85,56,100,41,115,26,70)(12,84,57,99,42,114,27,69)(13,83,58,98,43,113,28,68)(14,82,59,97,44,112,29,67)(15,81,60,96,45,111,30,66), (1,65,46,110,31,95,16,80)(2,114,47,99,32,84,17,69)(3,103,48,88,33,73,18,118)(4,92,49,77,34,62,19,107)(5,81,50,66,35,111,20,96)(6,70,51,115,36,100,21,85)(7,119,52,104,37,89,22,74)(8,108,53,93,38,78,23,63)(9,97,54,82,39,67,24,112)(10,86,55,71,40,116,25,101)(11,75,56,120,41,105,26,90)(12,64,57,109,42,94,27,79)(13,113,58,98,43,83,28,68)(14,102,59,87,44,72,29,117)(15,91,60,76,45,61,30,106) );
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)], [(1,95,46,110,31,65,16,80),(2,94,47,109,32,64,17,79),(3,93,48,108,33,63,18,78),(4,92,49,107,34,62,19,77),(5,91,50,106,35,61,20,76),(6,90,51,105,36,120,21,75),(7,89,52,104,37,119,22,74),(8,88,53,103,38,118,23,73),(9,87,54,102,39,117,24,72),(10,86,55,101,40,116,25,71),(11,85,56,100,41,115,26,70),(12,84,57,99,42,114,27,69),(13,83,58,98,43,113,28,68),(14,82,59,97,44,112,29,67),(15,81,60,96,45,111,30,66)], [(1,65,46,110,31,95,16,80),(2,114,47,99,32,84,17,69),(3,103,48,88,33,73,18,118),(4,92,49,77,34,62,19,107),(5,81,50,66,35,111,20,96),(6,70,51,115,36,100,21,85),(7,119,52,104,37,89,22,74),(8,108,53,93,38,78,23,63),(9,97,54,82,39,67,24,112),(10,86,55,71,40,116,25,101),(11,75,56,120,41,105,26,90),(12,64,57,109,42,94,27,79),(13,113,58,98,43,83,28,68),(14,102,59,87,44,72,29,117),(15,91,60,76,45,61,30,106)])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 60 | 60 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 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 | D10 | Dic5 | D12 | C3⋊D4 | C4×S3 | C5⋊D4 | C4.D4 | S3×D5 | M4(2)⋊S3 | C15⋊D4 | C5⋊D12 | S3×Dic5 | C20.D4 | C60.53D4 |
| kernel | C60.53D4 | C3×C4.Dic5 | C60.7C4 | C10×D12 | S3×C2×C10 | C4.Dic5 | C60 | C2×D12 | C2×C20 | C2×C12 | C22×S3 | C20 | C20 | C2×C10 | C12 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C60.53D4 ►in GL8(𝔽241)
| 205 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 24 | 87 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 226 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 191 | 0 | 0 |
| 0 | 0 | 0 | 0 | 121 | 187 | 0 | 0 |
| 0 | 0 | 0 | 0 | 192 | 73 | 185 | 144 |
| 0 | 0 | 0 | 0 | 184 | 99 | 87 | 56 |
| 18 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 133 | 223 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 202 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 149 | 0 | 77 | 240 |
| 0 | 0 | 0 | 0 | 234 | 144 | 39 | 0 |
| 0 | 0 | 0 | 0 | 36 | 56 | 76 | 0 |
| 223 | 209 | 0 | 0 | 0 | 0 | 0 | 0 |
| 108 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 202 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 162 | 0 | 103 | 1 |
| 0 | 0 | 0 | 0 | 234 | 144 | 39 | 0 |
| 0 | 0 | 0 | 0 | 171 | 56 | 76 | 0 |
G:=sub<GL(8,GF(241))| [205,24,0,0,0,0,0,0,0,87,0,0,0,0,0,0,0,0,226,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,54,121,192,184,0,0,0,0,191,187,73,99,0,0,0,0,0,0,185,87,0,0,0,0,0,0,144,56],[18,133,0,0,0,0,0,0,32,223,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,0,202,149,234,36,0,0,0,0,0,0,144,56,0,0,0,0,3,77,39,76,0,0,0,0,0,240,0,0],[223,108,0,0,0,0,0,0,209,18,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,202,162,234,171,0,0,0,0,0,0,144,56,0,0,0,0,3,103,39,76,0,0,0,0,0,1,0,0] >;
C60.53D4 in GAP, Magma, Sage, TeX
C_{60}._{53}D_4 % in TeX
G:=Group("C60.53D4"); // GroupNames label
G:=SmallGroup(480,35);
// by ID
G=gap.SmallGroup(480,35);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,100,675,1356,18822]);
// 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^49,c*b*c^-1=a^15*b^3>;
// generators/relations