metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80.5C4, C5⋊1M6(2), C16.3F5, D10.3C16, Dic5.3C16, C5⋊C32⋊1C2, C5⋊2C8.6C8, (C8×D5).8C4, (C4×D5).6C8, C8.36(C2×F5), C20.17(C2×C8), C10.2(C2×C16), C40.31(C2×C4), (D5×C16).8C2, C2.3(D5⋊C16), C4.12(D5⋊C8), C5⋊2C16.14C22, SmallGroup(320,180)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C80.C4
G = < a,b | a80=1, b4=a50, bab-1=a73 >
Smallest permutation representation of C80.C4
►On 160 points
Generators in S160
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 97 26 82 51 147 76 132 21 117 46 102 71 87 16 152 41 137 66 122 11 107 36 92 61 157 6 142 31 127 56 112)(2 154 75 155 52 124 45 125 22 94 15 95 72 144 65 145 42 114 35 115 12 84 5 85 62 134 55 135 32 104 25 105)(3 131 44 148 53 101 14 118 23 151 64 88 73 121 34 138 43 91 4 108 13 141 54 158 63 111 24 128 33 81 74 98)(7 119 80 120 57 89 50 90 27 139 20 140 77 109 70 110 47 159 40 160 17 129 10 130 67 99 60 100 37 149 30 150)(8 96 49 113 58 146 19 83 28 116 69 133 78 86 39 103 48 136 9 153 18 106 59 123 68 156 29 93 38 126 79 143)
G:=sub<Sym(160)| (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,97,26,82,51,147,76,132,21,117,46,102,71,87,16,152,41,137,66,122,11,107,36,92,61,157,6,142,31,127,56,112)(2,154,75,155,52,124,45,125,22,94,15,95,72,144,65,145,42,114,35,115,12,84,5,85,62,134,55,135,32,104,25,105)(3,131,44,148,53,101,14,118,23,151,64,88,73,121,34,138,43,91,4,108,13,141,54,158,63,111,24,128,33,81,74,98)(7,119,80,120,57,89,50,90,27,139,20,140,77,109,70,110,47,159,40,160,17,129,10,130,67,99,60,100,37,149,30,150)(8,96,49,113,58,146,19,83,28,116,69,133,78,86,39,103,48,136,9,153,18,106,59,123,68,156,29,93,38,126,79,143)>;
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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,97,26,82,51,147,76,132,21,117,46,102,71,87,16,152,41,137,66,122,11,107,36,92,61,157,6,142,31,127,56,112)(2,154,75,155,52,124,45,125,22,94,15,95,72,144,65,145,42,114,35,115,12,84,5,85,62,134,55,135,32,104,25,105)(3,131,44,148,53,101,14,118,23,151,64,88,73,121,34,138,43,91,4,108,13,141,54,158,63,111,24,128,33,81,74,98)(7,119,80,120,57,89,50,90,27,139,20,140,77,109,70,110,47,159,40,160,17,129,10,130,67,99,60,100,37,149,30,150)(8,96,49,113,58,146,19,83,28,116,69,133,78,86,39,103,48,136,9,153,18,106,59,123,68,156,29,93,38,126,79,143) );
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),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,97,26,82,51,147,76,132,21,117,46,102,71,87,16,152,41,137,66,122,11,107,36,92,61,157,6,142,31,127,56,112),(2,154,75,155,52,124,45,125,22,94,15,95,72,144,65,145,42,114,35,115,12,84,5,85,62,134,55,135,32,104,25,105),(3,131,44,148,53,101,14,118,23,151,64,88,73,121,34,138,43,91,4,108,13,141,54,158,63,111,24,128,33,81,74,98),(7,119,80,120,57,89,50,90,27,139,20,140,77,109,70,110,47,159,40,160,17,129,10,130,67,99,60,100,37,149,30,150),(8,96,49,113,58,146,19,83,28,116,69,133,78,86,39,103,48,136,9,153,18,106,59,123,68,156,29,93,38,126,79,143)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 10 | 16A | 16B | 16C | 16D | 16E | ··· | 16L | 20A | 20B | 32A | ··· | 32P | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 16 | 16 | 16 | 16 | 16 | ··· | 16 | 20 | 20 | 32 | ··· | 32 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 10 | 1 | 1 | 10 | 4 | 1 | 1 | 1 | 1 | 10 | 10 | 4 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 4 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | M6(2) | F5 | C2×F5 | D5⋊C8 | D5⋊C16 | C80.C4 |
| kernel | C80.C4 | C5⋊C32 | D5×C16 | C80 | C8×D5 | C5⋊2C8 | C4×D5 | Dic5 | D10 | C5 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 4 | 8 |
Matrix representation of C80.C4 ►in GL4(𝔽641) generated by
| 488 | 244 | 488 | 0 |
| 0 | 488 | 244 | 488 |
| 153 | 153 | 0 | 397 |
| 244 | 397 | 397 | 244 |
| 322 | 3 | 432 | 212 |
| 429 | 209 | 638 | 319 |
| 429 | 110 | 432 | 220 |
| 322 | 110 | 531 | 319 |
G:=sub<GL(4,GF(641))| [488,0,153,244,244,488,153,397,488,244,0,397,0,488,397,244],[322,429,429,322,3,209,110,110,432,638,432,531,212,319,220,319] >;
C80.C4 in GAP, Magma, Sage, TeX
C_{80}.C_4 % in TeX
G:=Group("C80.C4"); // GroupNames label
G:=SmallGroup(320,180);
// by ID
G=gap.SmallGroup(320,180);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,477,64,58,80,102,6278,3156]);
// Polycyclic
G:=Group<a,b|a^80=1,b^4=a^50,b*a*b^-1=a^73>;
// generators/relations
Export