metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.5D12, C60.53D4, (C2×D12).5D5, (C2×C20).41D6, C4.Dic5⋊2S3, C60.7C4⋊2C2, (C10×D12).1C2, (C2×C12).42D10, C3⋊1(C20.D4), C20.5(C3⋊D4), C12.6(C5⋊D4), C15⋊3(C4.D4), C10.39(D6⋊C4), (C22×S3).Dic5, C4.12(C15⋊D4), C5⋊4(C12.46D4), C4.19(C5⋊D12), 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×C30).81(C2×C4), (C2×C10).70(C4×S3), (C3×C4.Dic5)⋊1C2, (C2×C6).1(C2×Dic5), SmallGroup(480,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.5D12
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, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, D4, C23, C10, C10, C12, D6, C2×C6, C15, M4(2), C2×D4, C20, C2×C10, C2×C10, C3⋊C8, C24, D12, C2×C12, C22×S3, C5×S3, C30, C30, C4.D4, C5⋊2C8, C2×C20, C5×D4, C22×C10, C4.Dic3, C3×M4(2), C2×D12, C60, S3×C10, C2×C30, C4.Dic5, C4.Dic5, D4×C10, C12.46D4, C3×C5⋊2C8, C15⋊3C8, C5×D12, C2×C60, S3×C2×C10, C20.D4, C3×C4.Dic5, C60.7C4, C10×D12, C20.5D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, Dic5, D10, C4×S3, D12, C3⋊D4, C4.D4, C2×Dic5, C5⋊D4, D6⋊C4, S3×D5, C23.D5, C12.46D4, S3×Dic5, C15⋊D4, C5⋊D12, C20.D4, D6⋊Dic5, C20.5D12
►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 76 46 91 31 106 16 61)(2 75 47 90 32 105 17 120)(3 74 48 89 33 104 18 119)(4 73 49 88 34 103 19 118)(5 72 50 87 35 102 20 117)(6 71 51 86 36 101 21 116)(7 70 52 85 37 100 22 115)(8 69 53 84 38 99 23 114)(9 68 54 83 39 98 24 113)(10 67 55 82 40 97 25 112)(11 66 56 81 41 96 26 111)(12 65 57 80 42 95 27 110)(13 64 58 79 43 94 28 109)(14 63 59 78 44 93 29 108)(15 62 60 77 45 92 30 107)
(1 106 46 91 31 76 16 61)(2 95 47 80 32 65 17 110)(3 84 48 69 33 114 18 99)(4 73 49 118 34 103 19 88)(5 62 50 107 35 92 20 77)(6 111 51 96 36 81 21 66)(7 100 52 85 37 70 22 115)(8 89 53 74 38 119 23 104)(9 78 54 63 39 108 24 93)(10 67 55 112 40 97 25 82)(11 116 56 101 41 86 26 71)(12 105 57 90 42 75 27 120)(13 94 58 79 43 64 28 109)(14 83 59 68 44 113 29 98)(15 72 60 117 45 102 30 87)
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), (1,76,46,91,31,106,16,61)(2,75,47,90,32,105,17,120)(3,74,48,89,33,104,18,119)(4,73,49,88,34,103,19,118)(5,72,50,87,35,102,20,117)(6,71,51,86,36,101,21,116)(7,70,52,85,37,100,22,115)(8,69,53,84,38,99,23,114)(9,68,54,83,39,98,24,113)(10,67,55,82,40,97,25,112)(11,66,56,81,41,96,26,111)(12,65,57,80,42,95,27,110)(13,64,58,79,43,94,28,109)(14,63,59,78,44,93,29,108)(15,62,60,77,45,92,30,107), (1,106,46,91,31,76,16,61)(2,95,47,80,32,65,17,110)(3,84,48,69,33,114,18,99)(4,73,49,118,34,103,19,88)(5,62,50,107,35,92,20,77)(6,111,51,96,36,81,21,66)(7,100,52,85,37,70,22,115)(8,89,53,74,38,119,23,104)(9,78,54,63,39,108,24,93)(10,67,55,112,40,97,25,82)(11,116,56,101,41,86,26,71)(12,105,57,90,42,75,27,120)(13,94,58,79,43,64,28,109)(14,83,59,68,44,113,29,98)(15,72,60,117,45,102,30,87)>;
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), (1,76,46,91,31,106,16,61)(2,75,47,90,32,105,17,120)(3,74,48,89,33,104,18,119)(4,73,49,88,34,103,19,118)(5,72,50,87,35,102,20,117)(6,71,51,86,36,101,21,116)(7,70,52,85,37,100,22,115)(8,69,53,84,38,99,23,114)(9,68,54,83,39,98,24,113)(10,67,55,82,40,97,25,112)(11,66,56,81,41,96,26,111)(12,65,57,80,42,95,27,110)(13,64,58,79,43,94,28,109)(14,63,59,78,44,93,29,108)(15,62,60,77,45,92,30,107), (1,106,46,91,31,76,16,61)(2,95,47,80,32,65,17,110)(3,84,48,69,33,114,18,99)(4,73,49,118,34,103,19,88)(5,62,50,107,35,92,20,77)(6,111,51,96,36,81,21,66)(7,100,52,85,37,70,22,115)(8,89,53,74,38,119,23,104)(9,78,54,63,39,108,24,93)(10,67,55,112,40,97,25,82)(11,116,56,101,41,86,26,71)(12,105,57,90,42,75,27,120)(13,94,58,79,43,64,28,109)(14,83,59,68,44,113,29,98)(15,72,60,117,45,102,30,87) );
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)], [(1,76,46,91,31,106,16,61),(2,75,47,90,32,105,17,120),(3,74,48,89,33,104,18,119),(4,73,49,88,34,103,19,118),(5,72,50,87,35,102,20,117),(6,71,51,86,36,101,21,116),(7,70,52,85,37,100,22,115),(8,69,53,84,38,99,23,114),(9,68,54,83,39,98,24,113),(10,67,55,82,40,97,25,112),(11,66,56,81,41,96,26,111),(12,65,57,80,42,95,27,110),(13,64,58,79,43,94,28,109),(14,63,59,78,44,93,29,108),(15,62,60,77,45,92,30,107)], [(1,106,46,91,31,76,16,61),(2,95,47,80,32,65,17,110),(3,84,48,69,33,114,18,99),(4,73,49,118,34,103,19,88),(5,62,50,107,35,92,20,77),(6,111,51,96,36,81,21,66),(7,100,52,85,37,70,22,115),(8,89,53,74,38,119,23,104),(9,78,54,63,39,108,24,93),(10,67,55,112,40,97,25,82),(11,116,56,101,41,86,26,71),(12,105,57,90,42,75,27,120),(13,94,58,79,43,64,28,109),(14,83,59,68,44,113,29,98),(15,72,60,117,45,102,30,87)]])
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 | C12.46D4 | C15⋊D4 | C5⋊D12 | S3×Dic5 | C20.D4 | C20.5D12 |
| kernel | C20.5D12 | 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 C20.5D12 ►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] >;
C20.5D12 in GAP, Magma, Sage, TeX
C_{20}._5D_{12} % in TeX
G:=Group("C20.5D12"); // 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