metabelian, supersoluble, monomial, A-group
Aliases: C20.10F5, C5⋊D5⋊4C8, C52⋊7(C2×C8), C5⋊1(D5⋊C8), (C5×C20).11C4, C52⋊4C8⋊5C2, C10.14(C2×F5), C4.3(C5⋊F5), C52⋊6C4.18C22, (C2×C5⋊D5).7C4, (C4×C5⋊D5).10C2, C2.1(C2×C5⋊F5), (C5×C10).27(C2×C4), SmallGroup(400,149)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C20.F5 |
Generators and relations for C20.F5
G = < a,b,c | a20=b5=1, c4=a10, ab=ba, cac-1=a13, cbc-1=b3 >
Subgroups: 472 in 88 conjugacy classes, 32 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, D5, C10, C2×C8, Dic5, C20, D10, C52, C5⋊C8, C4×D5, C5⋊D5, C5×C10, D5⋊C8, C52⋊6C4, C5×C20, C2×C5⋊D5, C52⋊4C8, C4×C5⋊D5, C20.F5
Quotients: C1, C2, C4, C22, C8, C2×C4, C2×C8, F5, C2×F5, D5⋊C8, C5⋊F5, C2×C5⋊F5, C20.F5
►On 200 points
Generators in S200
(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)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)
(1 78 36 102 147)(2 79 37 103 148)(3 80 38 104 149)(4 61 39 105 150)(5 62 40 106 151)(6 63 21 107 152)(7 64 22 108 153)(8 65 23 109 154)(9 66 24 110 155)(10 67 25 111 156)(11 68 26 112 157)(12 69 27 113 158)(13 70 28 114 159)(14 71 29 115 160)(15 72 30 116 141)(16 73 31 117 142)(17 74 32 118 143)(18 75 33 119 144)(19 76 34 120 145)(20 77 35 101 146)(41 87 173 197 140)(42 88 174 198 121)(43 89 175 199 122)(44 90 176 200 123)(45 91 177 181 124)(46 92 178 182 125)(47 93 179 183 126)(48 94 180 184 127)(49 95 161 185 128)(50 96 162 186 129)(51 97 163 187 130)(52 98 164 188 131)(53 99 165 189 132)(54 100 166 190 133)(55 81 167 191 134)(56 82 168 192 135)(57 83 169 193 136)(58 84 170 194 137)(59 85 171 195 138)(60 86 172 196 139)
(1 192 6 197 11 182 16 187)(2 189 15 190 12 199 5 200)(3 186 4 183 13 196 14 193)(7 194 20 195 17 184 10 185)(8 191 9 188 18 181 19 198)(21 173 112 125 31 163 102 135)(22 170 101 138 32 180 111 128)(23 167 110 131 33 177 120 121)(24 164 119 124 34 174 109 134)(25 161 108 137 35 171 118 127)(26 178 117 130 36 168 107 140)(27 175 106 123 37 165 116 133)(28 172 115 136 38 162 105 126)(29 169 104 129 39 179 114 139)(30 166 113 122 40 176 103 132)(41 157 92 73 51 147 82 63)(42 154 81 66 52 144 91 76)(43 151 90 79 53 141 100 69)(44 148 99 72 54 158 89 62)(45 145 88 65 55 155 98 75)(46 142 97 78 56 152 87 68)(47 159 86 71 57 149 96 61)(48 156 95 64 58 146 85 74)(49 153 84 77 59 143 94 67)(50 150 93 70 60 160 83 80)
G:=sub<Sym(200)| (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200), (1,78,36,102,147)(2,79,37,103,148)(3,80,38,104,149)(4,61,39,105,150)(5,62,40,106,151)(6,63,21,107,152)(7,64,22,108,153)(8,65,23,109,154)(9,66,24,110,155)(10,67,25,111,156)(11,68,26,112,157)(12,69,27,113,158)(13,70,28,114,159)(14,71,29,115,160)(15,72,30,116,141)(16,73,31,117,142)(17,74,32,118,143)(18,75,33,119,144)(19,76,34,120,145)(20,77,35,101,146)(41,87,173,197,140)(42,88,174,198,121)(43,89,175,199,122)(44,90,176,200,123)(45,91,177,181,124)(46,92,178,182,125)(47,93,179,183,126)(48,94,180,184,127)(49,95,161,185,128)(50,96,162,186,129)(51,97,163,187,130)(52,98,164,188,131)(53,99,165,189,132)(54,100,166,190,133)(55,81,167,191,134)(56,82,168,192,135)(57,83,169,193,136)(58,84,170,194,137)(59,85,171,195,138)(60,86,172,196,139), (1,192,6,197,11,182,16,187)(2,189,15,190,12,199,5,200)(3,186,4,183,13,196,14,193)(7,194,20,195,17,184,10,185)(8,191,9,188,18,181,19,198)(21,173,112,125,31,163,102,135)(22,170,101,138,32,180,111,128)(23,167,110,131,33,177,120,121)(24,164,119,124,34,174,109,134)(25,161,108,137,35,171,118,127)(26,178,117,130,36,168,107,140)(27,175,106,123,37,165,116,133)(28,172,115,136,38,162,105,126)(29,169,104,129,39,179,114,139)(30,166,113,122,40,176,103,132)(41,157,92,73,51,147,82,63)(42,154,81,66,52,144,91,76)(43,151,90,79,53,141,100,69)(44,148,99,72,54,158,89,62)(45,145,88,65,55,155,98,75)(46,142,97,78,56,152,87,68)(47,159,86,71,57,149,96,61)(48,156,95,64,58,146,85,74)(49,153,84,77,59,143,94,67)(50,150,93,70,60,160,83,80)>;
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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200), (1,78,36,102,147)(2,79,37,103,148)(3,80,38,104,149)(4,61,39,105,150)(5,62,40,106,151)(6,63,21,107,152)(7,64,22,108,153)(8,65,23,109,154)(9,66,24,110,155)(10,67,25,111,156)(11,68,26,112,157)(12,69,27,113,158)(13,70,28,114,159)(14,71,29,115,160)(15,72,30,116,141)(16,73,31,117,142)(17,74,32,118,143)(18,75,33,119,144)(19,76,34,120,145)(20,77,35,101,146)(41,87,173,197,140)(42,88,174,198,121)(43,89,175,199,122)(44,90,176,200,123)(45,91,177,181,124)(46,92,178,182,125)(47,93,179,183,126)(48,94,180,184,127)(49,95,161,185,128)(50,96,162,186,129)(51,97,163,187,130)(52,98,164,188,131)(53,99,165,189,132)(54,100,166,190,133)(55,81,167,191,134)(56,82,168,192,135)(57,83,169,193,136)(58,84,170,194,137)(59,85,171,195,138)(60,86,172,196,139), (1,192,6,197,11,182,16,187)(2,189,15,190,12,199,5,200)(3,186,4,183,13,196,14,193)(7,194,20,195,17,184,10,185)(8,191,9,188,18,181,19,198)(21,173,112,125,31,163,102,135)(22,170,101,138,32,180,111,128)(23,167,110,131,33,177,120,121)(24,164,119,124,34,174,109,134)(25,161,108,137,35,171,118,127)(26,178,117,130,36,168,107,140)(27,175,106,123,37,165,116,133)(28,172,115,136,38,162,105,126)(29,169,104,129,39,179,114,139)(30,166,113,122,40,176,103,132)(41,157,92,73,51,147,82,63)(42,154,81,66,52,144,91,76)(43,151,90,79,53,141,100,69)(44,148,99,72,54,158,89,62)(45,145,88,65,55,155,98,75)(46,142,97,78,56,152,87,68)(47,159,86,71,57,149,96,61)(48,156,95,64,58,146,85,74)(49,153,84,77,59,143,94,67)(50,150,93,70,60,160,83,80) );
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),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)], [(1,78,36,102,147),(2,79,37,103,148),(3,80,38,104,149),(4,61,39,105,150),(5,62,40,106,151),(6,63,21,107,152),(7,64,22,108,153),(8,65,23,109,154),(9,66,24,110,155),(10,67,25,111,156),(11,68,26,112,157),(12,69,27,113,158),(13,70,28,114,159),(14,71,29,115,160),(15,72,30,116,141),(16,73,31,117,142),(17,74,32,118,143),(18,75,33,119,144),(19,76,34,120,145),(20,77,35,101,146),(41,87,173,197,140),(42,88,174,198,121),(43,89,175,199,122),(44,90,176,200,123),(45,91,177,181,124),(46,92,178,182,125),(47,93,179,183,126),(48,94,180,184,127),(49,95,161,185,128),(50,96,162,186,129),(51,97,163,187,130),(52,98,164,188,131),(53,99,165,189,132),(54,100,166,190,133),(55,81,167,191,134),(56,82,168,192,135),(57,83,169,193,136),(58,84,170,194,137),(59,85,171,195,138),(60,86,172,196,139)], [(1,192,6,197,11,182,16,187),(2,189,15,190,12,199,5,200),(3,186,4,183,13,196,14,193),(7,194,20,195,17,184,10,185),(8,191,9,188,18,181,19,198),(21,173,112,125,31,163,102,135),(22,170,101,138,32,180,111,128),(23,167,110,131,33,177,120,121),(24,164,119,124,34,174,109,134),(25,161,108,137,35,171,118,127),(26,178,117,130,36,168,107,140),(27,175,106,123,37,165,116,133),(28,172,115,136,38,162,105,126),(29,169,104,129,39,179,114,139),(30,166,113,122,40,176,103,132),(41,157,92,73,51,147,82,63),(42,154,81,66,52,144,91,76),(43,151,90,79,53,141,100,69),(44,148,99,72,54,158,89,62),(45,145,88,65,55,155,98,75),(46,142,97,78,56,152,87,68),(47,159,86,71,57,149,96,61),(48,156,95,64,58,146,85,74),(49,153,84,77,59,143,94,67),(50,150,93,70,60,160,83,80)]])
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | ··· | 5F | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | ··· | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 25 | 25 | 1 | 1 | 25 | 25 | 4 | ··· | 4 | 25 | ··· | 25 | 4 | ··· | 4 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | F5 | C2×F5 | D5⋊C8 |
| kernel | C20.F5 | C52⋊4C8 | C4×C5⋊D5 | C5×C20 | C2×C5⋊D5 | C5⋊D5 | C20 | C10 | C5 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 6 | 6 | 12 |
Matrix representation of C20.F5 ►in GL8(𝔽41)
| 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 32 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 40 |
| 14 | 27 | 14 | 0 | 0 | 0 | 0 | 0 |
| 28 | 27 | 0 | 14 | 0 | 0 | 0 | 0 |
| 14 | 0 | 27 | 28 | 0 | 0 | 0 | 0 |
| 0 | 14 | 27 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 28 | 27 | 0 |
| 0 | 0 | 0 | 0 | 40 | 28 | 0 | 13 |
| 0 | 0 | 0 | 0 | 13 | 0 | 28 | 40 |
| 0 | 0 | 0 | 0 | 0 | 27 | 28 | 13 |
G:=sub<GL(8,GF(41))| [0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,9,9,9,9,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,32,32,32,32],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[14,28,14,0,0,0,0,0,27,27,0,14,0,0,0,0,14,0,27,27,0,0,0,0,0,14,28,14,0,0,0,0,0,0,0,0,13,40,13,0,0,0,0,0,28,28,0,27,0,0,0,0,27,0,28,28,0,0,0,0,0,13,40,13] >;
C20.F5 in GAP, Magma, Sage, TeX
C_{20}.F_5 % in TeX
G:=Group("C20.F5"); // GroupNames label
G:=SmallGroup(400,149);
// by ID
G=gap.SmallGroup(400,149);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,50,964,496,5765,2897]);
// Polycyclic
G:=Group<a,b,c|a^20=b^5=1,c^4=a^10,a*b=b*a,c*a*c^-1=a^13,c*b*c^-1=b^3>;
// generators/relations