metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C60.3C4, C12.3F5, C20.3Dic3, D10.2Dic3, Dic5.15D6, D5⋊(C3⋊C8), C15⋊3(C2×C8), C3⋊2(D5⋊C8), (C3×D5)⋊2C8, C15⋊C8⋊7C2, C4.3(C3⋊F5), C6.8(C2×F5), C30.8(C2×C4), (C6×D5).4C4, (C4×D5).5S3, (D5×C12).8C2, C10.1(C2×Dic3), (C3×Dic5).21C22, C5⋊1(C2×C3⋊C8), C2.1(C2×C3⋊F5), SmallGroup(240,118)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C60.C4 |
Generators and relations for C60.C4
G = < a,b | a60=1, b4=a30, bab-1=a17 >
Smallest permutation representation of C60.C4
►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 61 16 76 31 91 46 106)(2 114 5 93 32 84 35 63)(3 107 54 110 33 77 24 80)(4 100 43 67 34 70 13 97)(6 86 21 101 36 116 51 71)(7 79 10 118 37 109 40 88)(8 72 59 75 38 102 29 105)(9 65 48 92 39 95 18 62)(11 111 26 66 41 81 56 96)(12 104 15 83 42 74 45 113)(14 90 53 117 44 120 23 87)(17 69 20 108 47 99 50 78)(19 115 58 82 49 85 28 112)(22 94 25 73 52 64 55 103)(27 119 30 98 57 89 60 68)
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), (1,61,16,76,31,91,46,106)(2,114,5,93,32,84,35,63)(3,107,54,110,33,77,24,80)(4,100,43,67,34,70,13,97)(6,86,21,101,36,116,51,71)(7,79,10,118,37,109,40,88)(8,72,59,75,38,102,29,105)(9,65,48,92,39,95,18,62)(11,111,26,66,41,81,56,96)(12,104,15,83,42,74,45,113)(14,90,53,117,44,120,23,87)(17,69,20,108,47,99,50,78)(19,115,58,82,49,85,28,112)(22,94,25,73,52,64,55,103)(27,119,30,98,57,89,60,68)>;
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), (1,61,16,76,31,91,46,106)(2,114,5,93,32,84,35,63)(3,107,54,110,33,77,24,80)(4,100,43,67,34,70,13,97)(6,86,21,101,36,116,51,71)(7,79,10,118,37,109,40,88)(8,72,59,75,38,102,29,105)(9,65,48,92,39,95,18,62)(11,111,26,66,41,81,56,96)(12,104,15,83,42,74,45,113)(14,90,53,117,44,120,23,87)(17,69,20,108,47,99,50,78)(19,115,58,82,49,85,28,112)(22,94,25,73,52,64,55,103)(27,119,30,98,57,89,60,68) );
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)], [(1,61,16,76,31,91,46,106),(2,114,5,93,32,84,35,63),(3,107,54,110,33,77,24,80),(4,100,43,67,34,70,13,97),(6,86,21,101,36,116,51,71),(7,79,10,118,37,109,40,88),(8,72,59,75,38,102,29,105),(9,65,48,92,39,95,18,62),(11,111,26,66,41,81,56,96),(12,104,15,83,42,74,45,113),(14,90,53,117,44,120,23,87),(17,69,20,108,47,99,50,78),(19,115,58,82,49,85,28,112),(22,94,25,73,52,64,55,103),(27,119,30,98,57,89,60,68)]])
C60.C4 is a maximal subgroup of
F5×C3⋊C8 C30.3C42 D60⋊C4 Dic6⋊F5 C8×C3⋊F5 C24⋊F5 D20⋊Dic3 Dic10⋊2Dic3 S3×D5⋊C8 D12.F5 Dic6.F5 C5⋊C8⋊D6 C60.59(C2×C4) Dic10.Dic3 D20.Dic3
C60.C4 is a maximal quotient of
C24.F5 C120.C4 C4×C15⋊C8 C30.7M4(2) Dic5.13D12
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 6C | 8A | ··· | 8H | 10 | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 2 | 1 | 1 | 5 | 5 | 4 | 2 | 10 | 10 | 15 | ··· | 15 | 4 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | D6 | Dic3 | Dic3 | C3⋊C8 | F5 | C2×F5 | C3⋊F5 | D5⋊C8 | C2×C3⋊F5 | C60.C4 |
| kernel | C60.C4 | C15⋊C8 | D5×C12 | C60 | C6×D5 | C3×D5 | C4×D5 | Dic5 | C20 | D10 | D5 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C60.C4 ►in GL6(𝔽241)
| 0 | 64 | 0 | 0 | 0 | 0 |
| 177 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 127 |
| 0 | 0 | 114 | 126 | 126 | 114 |
| 0 | 0 | 127 | 0 | 12 | 12 |
| 0 | 0 | 229 | 115 | 229 | 0 |
| 181 | 227 | 0 | 0 | 0 | 0 |
| 46 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 0 | 59 | 59 |
| 0 | 0 | 0 | 182 | 208 | 182 |
| 0 | 0 | 215 | 33 | 33 | 215 |
| 0 | 0 | 182 | 208 | 182 | 0 |
G:=sub<GL(6,GF(241))| [0,177,0,0,0,0,64,177,0,0,0,0,0,0,12,114,127,229,0,0,12,126,0,115,0,0,0,126,12,229,0,0,127,114,12,0],[181,46,0,0,0,0,227,60,0,0,0,0,0,0,26,0,215,182,0,0,0,182,33,208,0,0,59,208,33,182,0,0,59,182,215,0] >;
C60.C4 in GAP, Magma, Sage, TeX
C_{60}.C_4 % in TeX
G:=Group("C60.C4"); // GroupNames label
G:=SmallGroup(240,118);
// by ID
G=gap.SmallGroup(240,118);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,55,50,964,5189,1745]);
// Polycyclic
G:=Group<a,b|a^60=1,b^4=a^30,b*a*b^-1=a^17>;
// generators/relations
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