metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.8D20, C40.58D4, C42.263D10, (C4×C8)⋊9D5, (C4×C40)⋊14C2, (C2×D40).2C2, C4.32(C2×D20), (C2×C4).62D20, (C2×Dic20)⋊1C2, C10.4(C4○D8), (C2×C20).352D4, C20.275(C2×D4), (C2×C8).318D10, C4.D20⋊1C2, C5⋊1(C8.12D4), C2.7(C20⋊4D4), C10.5(C4⋊1D4), (C2×D20).6C22, C22.92(C2×D20), C2.7(D40⋊7C2), (C2×C40).390C22, (C4×C20).309C22, (C2×C20).725C23, (C2×Dic10).5C22, (C2×C40⋊C2)⋊8C2, (C2×C10).108(C2×D4), (C2×C4).668(C22×D5), SmallGroup(320,323)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.8D20
G = < a,b,c | a8=b20=1, c2=a4, ab=ba, cac-1=a-1, cbc-1=a4b-1 >
Subgroups: 686 in 130 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×Q16, C40, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C8.12D4, C40⋊C2, D40, Dic20, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×D20, C4×C40, C4.D20, C2×C40⋊C2, C2×D40, C2×Dic20, C8.8D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C4○D8, D20, C22×D5, C8.12D4, C2×D20, C20⋊4D4, D40⋊7C2, C8.8D20
►On 160 points
Generators in S160
(1 93 35 62 139 52 106 146)(2 94 36 63 140 53 107 147)(3 95 37 64 121 54 108 148)(4 96 38 65 122 55 109 149)(5 97 39 66 123 56 110 150)(6 98 40 67 124 57 111 151)(7 99 21 68 125 58 112 152)(8 100 22 69 126 59 113 153)(9 81 23 70 127 60 114 154)(10 82 24 71 128 41 115 155)(11 83 25 72 129 42 116 156)(12 84 26 73 130 43 117 157)(13 85 27 74 131 44 118 158)(14 86 28 75 132 45 119 159)(15 87 29 76 133 46 120 160)(16 88 30 77 134 47 101 141)(17 89 31 78 135 48 102 142)(18 90 32 79 136 49 103 143)(19 91 33 80 137 50 104 144)(20 92 34 61 138 51 105 145)
(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)
(1 138 139 20)(2 19 140 137)(3 136 121 18)(4 17 122 135)(5 134 123 16)(6 15 124 133)(7 132 125 14)(8 13 126 131)(9 130 127 12)(10 11 128 129)(21 28 112 119)(22 118 113 27)(23 26 114 117)(24 116 115 25)(29 40 120 111)(30 110 101 39)(31 38 102 109)(32 108 103 37)(33 36 104 107)(34 106 105 35)(41 72 82 156)(42 155 83 71)(43 70 84 154)(44 153 85 69)(45 68 86 152)(46 151 87 67)(47 66 88 150)(48 149 89 65)(49 64 90 148)(50 147 91 63)(51 62 92 146)(52 145 93 61)(53 80 94 144)(54 143 95 79)(55 78 96 142)(56 141 97 77)(57 76 98 160)(58 159 99 75)(59 74 100 158)(60 157 81 73)
G:=sub<Sym(160)| (1,93,35,62,139,52,106,146)(2,94,36,63,140,53,107,147)(3,95,37,64,121,54,108,148)(4,96,38,65,122,55,109,149)(5,97,39,66,123,56,110,150)(6,98,40,67,124,57,111,151)(7,99,21,68,125,58,112,152)(8,100,22,69,126,59,113,153)(9,81,23,70,127,60,114,154)(10,82,24,71,128,41,115,155)(11,83,25,72,129,42,116,156)(12,84,26,73,130,43,117,157)(13,85,27,74,131,44,118,158)(14,86,28,75,132,45,119,159)(15,87,29,76,133,46,120,160)(16,88,30,77,134,47,101,141)(17,89,31,78,135,48,102,142)(18,90,32,79,136,49,103,143)(19,91,33,80,137,50,104,144)(20,92,34,61,138,51,105,145), (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), (1,138,139,20)(2,19,140,137)(3,136,121,18)(4,17,122,135)(5,134,123,16)(6,15,124,133)(7,132,125,14)(8,13,126,131)(9,130,127,12)(10,11,128,129)(21,28,112,119)(22,118,113,27)(23,26,114,117)(24,116,115,25)(29,40,120,111)(30,110,101,39)(31,38,102,109)(32,108,103,37)(33,36,104,107)(34,106,105,35)(41,72,82,156)(42,155,83,71)(43,70,84,154)(44,153,85,69)(45,68,86,152)(46,151,87,67)(47,66,88,150)(48,149,89,65)(49,64,90,148)(50,147,91,63)(51,62,92,146)(52,145,93,61)(53,80,94,144)(54,143,95,79)(55,78,96,142)(56,141,97,77)(57,76,98,160)(58,159,99,75)(59,74,100,158)(60,157,81,73)>;
G:=Group( (1,93,35,62,139,52,106,146)(2,94,36,63,140,53,107,147)(3,95,37,64,121,54,108,148)(4,96,38,65,122,55,109,149)(5,97,39,66,123,56,110,150)(6,98,40,67,124,57,111,151)(7,99,21,68,125,58,112,152)(8,100,22,69,126,59,113,153)(9,81,23,70,127,60,114,154)(10,82,24,71,128,41,115,155)(11,83,25,72,129,42,116,156)(12,84,26,73,130,43,117,157)(13,85,27,74,131,44,118,158)(14,86,28,75,132,45,119,159)(15,87,29,76,133,46,120,160)(16,88,30,77,134,47,101,141)(17,89,31,78,135,48,102,142)(18,90,32,79,136,49,103,143)(19,91,33,80,137,50,104,144)(20,92,34,61,138,51,105,145), (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), (1,138,139,20)(2,19,140,137)(3,136,121,18)(4,17,122,135)(5,134,123,16)(6,15,124,133)(7,132,125,14)(8,13,126,131)(9,130,127,12)(10,11,128,129)(21,28,112,119)(22,118,113,27)(23,26,114,117)(24,116,115,25)(29,40,120,111)(30,110,101,39)(31,38,102,109)(32,108,103,37)(33,36,104,107)(34,106,105,35)(41,72,82,156)(42,155,83,71)(43,70,84,154)(44,153,85,69)(45,68,86,152)(46,151,87,67)(47,66,88,150)(48,149,89,65)(49,64,90,148)(50,147,91,63)(51,62,92,146)(52,145,93,61)(53,80,94,144)(54,143,95,79)(55,78,96,142)(56,141,97,77)(57,76,98,160)(58,159,99,75)(59,74,100,158)(60,157,81,73) );
G=PermutationGroup([[(1,93,35,62,139,52,106,146),(2,94,36,63,140,53,107,147),(3,95,37,64,121,54,108,148),(4,96,38,65,122,55,109,149),(5,97,39,66,123,56,110,150),(6,98,40,67,124,57,111,151),(7,99,21,68,125,58,112,152),(8,100,22,69,126,59,113,153),(9,81,23,70,127,60,114,154),(10,82,24,71,128,41,115,155),(11,83,25,72,129,42,116,156),(12,84,26,73,130,43,117,157),(13,85,27,74,131,44,118,158),(14,86,28,75,132,45,119,159),(15,87,29,76,133,46,120,160),(16,88,30,77,134,47,101,141),(17,89,31,78,135,48,102,142),(18,90,32,79,136,49,103,143),(19,91,33,80,137,50,104,144),(20,92,34,61,138,51,105,145)], [(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)], [(1,138,139,20),(2,19,140,137),(3,136,121,18),(4,17,122,135),(5,134,123,16),(6,15,124,133),(7,132,125,14),(8,13,126,131),(9,130,127,12),(10,11,128,129),(21,28,112,119),(22,118,113,27),(23,26,114,117),(24,116,115,25),(29,40,120,111),(30,110,101,39),(31,38,102,109),(32,108,103,37),(33,36,104,107),(34,106,105,35),(41,72,82,156),(42,155,83,71),(43,70,84,154),(44,153,85,69),(45,68,86,152),(46,151,87,67),(47,66,88,150),(48,149,89,65),(49,64,90,148),(50,147,91,63),(51,62,92,146),(52,145,93,61),(53,80,94,144),(54,143,95,79),(55,78,96,142),(56,141,97,77),(57,76,98,160),(58,159,99,75),(59,74,100,158),(60,157,81,73)]])
86 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 40 | 2 | ··· | 2 | 40 | 40 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | C4○D8 | D20 | D20 | D40⋊7C2 |
| kernel | C8.8D20 | C4×C40 | C4.D20 | C2×C40⋊C2 | C2×D40 | C2×Dic20 | C40 | C2×C20 | C4×C8 | C42 | C2×C8 | C10 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 16 | 8 | 32 |
Matrix representation of C8.8D20 ►in GL4(𝔽41) generated by
| 12 | 8 | 0 | 0 |
| 33 | 5 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 13 | 9 | 0 | 0 |
| 32 | 0 | 0 | 0 |
| 0 | 0 | 32 | 30 |
| 0 | 0 | 11 | 27 |
| 9 | 13 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 30 |
| 0 | 0 | 11 | 9 |
G:=sub<GL(4,GF(41))| [12,33,0,0,8,5,0,0,0,0,40,0,0,0,0,40],[13,32,0,0,9,0,0,0,0,0,32,11,0,0,30,27],[9,0,0,0,13,32,0,0,0,0,32,11,0,0,30,9] >;
C8.8D20 in GAP, Magma, Sage, TeX
C_8._8D_{20} % in TeX
G:=Group("C8.8D20"); // GroupNames label
G:=SmallGroup(320,323);
// by ID
G=gap.SmallGroup(320,323);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=b^20=1,c^2=a^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations