metabelian, supersoluble, monomial
Aliases: C5⋊1D40, C40⋊1D5, C52⋊5D8, C10.8D20, C20.47D10, (C5×C40)⋊1C2, C8⋊1(C5⋊D5), C20⋊D5⋊1C2, (C5×C10).23D4, C2.4(C20⋊D5), (C5×C20).33C22, C4.9(C2×C5⋊D5), SmallGroup(400,95)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊5D8
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, ac=ca, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 872 in 88 conjugacy classes, 35 normal (9 characteristic)
C1, C2, C2, C4, C22, C5, C8, D4, D5, C10, D8, C20, D10, C52, C40, D20, C5⋊D5, C5×C10, D40, C5×C20, C2×C5⋊D5, C5×C40, C20⋊D5, C52⋊5D8
Quotients: C1, C2, C22, D4, D5, D8, D10, D20, C5⋊D5, D40, C2×C5⋊D5, C20⋊D5, C52⋊5D8
►On 200 points
Generators in S200
(1 69 116 189 107)(2 70 117 190 108)(3 71 118 191 109)(4 72 119 192 110)(5 65 120 185 111)(6 66 113 186 112)(7 67 114 187 105)(8 68 115 188 106)(9 171 140 35 93)(10 172 141 36 94)(11 173 142 37 95)(12 174 143 38 96)(13 175 144 39 89)(14 176 137 40 90)(15 169 138 33 91)(16 170 139 34 92)(17 62 99 181 53)(18 63 100 182 54)(19 64 101 183 55)(20 57 102 184 56)(21 58 103 177 49)(22 59 104 178 50)(23 60 97 179 51)(24 61 98 180 52)(25 159 79 48 150)(26 160 80 41 151)(27 153 73 42 152)(28 154 74 43 145)(29 155 75 44 146)(30 156 76 45 147)(31 157 77 46 148)(32 158 78 47 149)(81 128 167 133 196)(82 121 168 134 197)(83 122 161 135 198)(84 123 162 136 199)(85 124 163 129 200)(86 125 164 130 193)(87 126 165 131 194)(88 127 166 132 195)
(1 44 168 89 100)(2 45 161 90 101)(3 46 162 91 102)(4 47 163 92 103)(5 48 164 93 104)(6 41 165 94 97)(7 42 166 95 98)(8 43 167 96 99)(9 178 65 150 130)(10 179 66 151 131)(11 180 67 152 132)(12 181 68 145 133)(13 182 69 146 134)(14 183 70 147 135)(15 184 71 148 136)(16 177 72 149 129)(17 188 154 81 143)(18 189 155 82 144)(19 190 156 83 137)(20 191 157 84 138)(21 192 158 85 139)(22 185 159 86 140)(23 186 160 87 141)(24 187 153 88 142)(25 193 171 50 120)(26 194 172 51 113)(27 195 173 52 114)(28 196 174 53 115)(29 197 175 54 116)(30 198 176 55 117)(31 199 169 56 118)(32 200 170 49 119)(33 57 109 77 123)(34 58 110 78 124)(35 59 111 79 125)(36 60 112 80 126)(37 61 105 73 127)(38 62 106 74 128)(39 63 107 75 121)(40 64 108 76 122)
(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 7)(2 6)(3 5)(9 123)(10 122)(11 121)(12 128)(13 127)(14 126)(15 125)(16 124)(17 28)(18 27)(19 26)(20 25)(21 32)(22 31)(23 30)(24 29)(33 130)(34 129)(35 136)(36 135)(37 134)(38 133)(39 132)(40 131)(41 101)(42 100)(43 99)(44 98)(45 97)(46 104)(47 103)(48 102)(49 158)(50 157)(51 156)(52 155)(53 154)(54 153)(55 160)(56 159)(57 150)(58 149)(59 148)(60 147)(61 146)(62 145)(63 152)(64 151)(65 109)(66 108)(67 107)(68 106)(69 105)(70 112)(71 111)(72 110)(73 182)(74 181)(75 180)(76 179)(77 178)(78 177)(79 184)(80 183)(81 174)(82 173)(83 172)(84 171)(85 170)(86 169)(87 176)(88 175)(89 166)(90 165)(91 164)(92 163)(93 162)(94 161)(95 168)(96 167)(113 190)(114 189)(115 188)(116 187)(117 186)(118 185)(119 192)(120 191)(137 194)(138 193)(139 200)(140 199)(141 198)(142 197)(143 196)(144 195)
G:=sub<Sym(200)| (1,69,116,189,107)(2,70,117,190,108)(3,71,118,191,109)(4,72,119,192,110)(5,65,120,185,111)(6,66,113,186,112)(7,67,114,187,105)(8,68,115,188,106)(9,171,140,35,93)(10,172,141,36,94)(11,173,142,37,95)(12,174,143,38,96)(13,175,144,39,89)(14,176,137,40,90)(15,169,138,33,91)(16,170,139,34,92)(17,62,99,181,53)(18,63,100,182,54)(19,64,101,183,55)(20,57,102,184,56)(21,58,103,177,49)(22,59,104,178,50)(23,60,97,179,51)(24,61,98,180,52)(25,159,79,48,150)(26,160,80,41,151)(27,153,73,42,152)(28,154,74,43,145)(29,155,75,44,146)(30,156,76,45,147)(31,157,77,46,148)(32,158,78,47,149)(81,128,167,133,196)(82,121,168,134,197)(83,122,161,135,198)(84,123,162,136,199)(85,124,163,129,200)(86,125,164,130,193)(87,126,165,131,194)(88,127,166,132,195), (1,44,168,89,100)(2,45,161,90,101)(3,46,162,91,102)(4,47,163,92,103)(5,48,164,93,104)(6,41,165,94,97)(7,42,166,95,98)(8,43,167,96,99)(9,178,65,150,130)(10,179,66,151,131)(11,180,67,152,132)(12,181,68,145,133)(13,182,69,146,134)(14,183,70,147,135)(15,184,71,148,136)(16,177,72,149,129)(17,188,154,81,143)(18,189,155,82,144)(19,190,156,83,137)(20,191,157,84,138)(21,192,158,85,139)(22,185,159,86,140)(23,186,160,87,141)(24,187,153,88,142)(25,193,171,50,120)(26,194,172,51,113)(27,195,173,52,114)(28,196,174,53,115)(29,197,175,54,116)(30,198,176,55,117)(31,199,169,56,118)(32,200,170,49,119)(33,57,109,77,123)(34,58,110,78,124)(35,59,111,79,125)(36,60,112,80,126)(37,61,105,73,127)(38,62,106,74,128)(39,63,107,75,121)(40,64,108,76,122), (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), (1,7)(2,6)(3,5)(9,123)(10,122)(11,121)(12,128)(13,127)(14,126)(15,125)(16,124)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)(33,130)(34,129)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,101)(42,100)(43,99)(44,98)(45,97)(46,104)(47,103)(48,102)(49,158)(50,157)(51,156)(52,155)(53,154)(54,153)(55,160)(56,159)(57,150)(58,149)(59,148)(60,147)(61,146)(62,145)(63,152)(64,151)(65,109)(66,108)(67,107)(68,106)(69,105)(70,112)(71,111)(72,110)(73,182)(74,181)(75,180)(76,179)(77,178)(78,177)(79,184)(80,183)(81,174)(82,173)(83,172)(84,171)(85,170)(86,169)(87,176)(88,175)(89,166)(90,165)(91,164)(92,163)(93,162)(94,161)(95,168)(96,167)(113,190)(114,189)(115,188)(116,187)(117,186)(118,185)(119,192)(120,191)(137,194)(138,193)(139,200)(140,199)(141,198)(142,197)(143,196)(144,195)>;
G:=Group( (1,69,116,189,107)(2,70,117,190,108)(3,71,118,191,109)(4,72,119,192,110)(5,65,120,185,111)(6,66,113,186,112)(7,67,114,187,105)(8,68,115,188,106)(9,171,140,35,93)(10,172,141,36,94)(11,173,142,37,95)(12,174,143,38,96)(13,175,144,39,89)(14,176,137,40,90)(15,169,138,33,91)(16,170,139,34,92)(17,62,99,181,53)(18,63,100,182,54)(19,64,101,183,55)(20,57,102,184,56)(21,58,103,177,49)(22,59,104,178,50)(23,60,97,179,51)(24,61,98,180,52)(25,159,79,48,150)(26,160,80,41,151)(27,153,73,42,152)(28,154,74,43,145)(29,155,75,44,146)(30,156,76,45,147)(31,157,77,46,148)(32,158,78,47,149)(81,128,167,133,196)(82,121,168,134,197)(83,122,161,135,198)(84,123,162,136,199)(85,124,163,129,200)(86,125,164,130,193)(87,126,165,131,194)(88,127,166,132,195), (1,44,168,89,100)(2,45,161,90,101)(3,46,162,91,102)(4,47,163,92,103)(5,48,164,93,104)(6,41,165,94,97)(7,42,166,95,98)(8,43,167,96,99)(9,178,65,150,130)(10,179,66,151,131)(11,180,67,152,132)(12,181,68,145,133)(13,182,69,146,134)(14,183,70,147,135)(15,184,71,148,136)(16,177,72,149,129)(17,188,154,81,143)(18,189,155,82,144)(19,190,156,83,137)(20,191,157,84,138)(21,192,158,85,139)(22,185,159,86,140)(23,186,160,87,141)(24,187,153,88,142)(25,193,171,50,120)(26,194,172,51,113)(27,195,173,52,114)(28,196,174,53,115)(29,197,175,54,116)(30,198,176,55,117)(31,199,169,56,118)(32,200,170,49,119)(33,57,109,77,123)(34,58,110,78,124)(35,59,111,79,125)(36,60,112,80,126)(37,61,105,73,127)(38,62,106,74,128)(39,63,107,75,121)(40,64,108,76,122), (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), (1,7)(2,6)(3,5)(9,123)(10,122)(11,121)(12,128)(13,127)(14,126)(15,125)(16,124)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)(33,130)(34,129)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,101)(42,100)(43,99)(44,98)(45,97)(46,104)(47,103)(48,102)(49,158)(50,157)(51,156)(52,155)(53,154)(54,153)(55,160)(56,159)(57,150)(58,149)(59,148)(60,147)(61,146)(62,145)(63,152)(64,151)(65,109)(66,108)(67,107)(68,106)(69,105)(70,112)(71,111)(72,110)(73,182)(74,181)(75,180)(76,179)(77,178)(78,177)(79,184)(80,183)(81,174)(82,173)(83,172)(84,171)(85,170)(86,169)(87,176)(88,175)(89,166)(90,165)(91,164)(92,163)(93,162)(94,161)(95,168)(96,167)(113,190)(114,189)(115,188)(116,187)(117,186)(118,185)(119,192)(120,191)(137,194)(138,193)(139,200)(140,199)(141,198)(142,197)(143,196)(144,195) );
G=PermutationGroup([[(1,69,116,189,107),(2,70,117,190,108),(3,71,118,191,109),(4,72,119,192,110),(5,65,120,185,111),(6,66,113,186,112),(7,67,114,187,105),(8,68,115,188,106),(9,171,140,35,93),(10,172,141,36,94),(11,173,142,37,95),(12,174,143,38,96),(13,175,144,39,89),(14,176,137,40,90),(15,169,138,33,91),(16,170,139,34,92),(17,62,99,181,53),(18,63,100,182,54),(19,64,101,183,55),(20,57,102,184,56),(21,58,103,177,49),(22,59,104,178,50),(23,60,97,179,51),(24,61,98,180,52),(25,159,79,48,150),(26,160,80,41,151),(27,153,73,42,152),(28,154,74,43,145),(29,155,75,44,146),(30,156,76,45,147),(31,157,77,46,148),(32,158,78,47,149),(81,128,167,133,196),(82,121,168,134,197),(83,122,161,135,198),(84,123,162,136,199),(85,124,163,129,200),(86,125,164,130,193),(87,126,165,131,194),(88,127,166,132,195)], [(1,44,168,89,100),(2,45,161,90,101),(3,46,162,91,102),(4,47,163,92,103),(5,48,164,93,104),(6,41,165,94,97),(7,42,166,95,98),(8,43,167,96,99),(9,178,65,150,130),(10,179,66,151,131),(11,180,67,152,132),(12,181,68,145,133),(13,182,69,146,134),(14,183,70,147,135),(15,184,71,148,136),(16,177,72,149,129),(17,188,154,81,143),(18,189,155,82,144),(19,190,156,83,137),(20,191,157,84,138),(21,192,158,85,139),(22,185,159,86,140),(23,186,160,87,141),(24,187,153,88,142),(25,193,171,50,120),(26,194,172,51,113),(27,195,173,52,114),(28,196,174,53,115),(29,197,175,54,116),(30,198,176,55,117),(31,199,169,56,118),(32,200,170,49,119),(33,57,109,77,123),(34,58,110,78,124),(35,59,111,79,125),(36,60,112,80,126),(37,61,105,73,127),(38,62,106,74,128),(39,63,107,75,121),(40,64,108,76,122)], [(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)], [(1,7),(2,6),(3,5),(9,123),(10,122),(11,121),(12,128),(13,127),(14,126),(15,125),(16,124),(17,28),(18,27),(19,26),(20,25),(21,32),(22,31),(23,30),(24,29),(33,130),(34,129),(35,136),(36,135),(37,134),(38,133),(39,132),(40,131),(41,101),(42,100),(43,99),(44,98),(45,97),(46,104),(47,103),(48,102),(49,158),(50,157),(51,156),(52,155),(53,154),(54,153),(55,160),(56,159),(57,150),(58,149),(59,148),(60,147),(61,146),(62,145),(63,152),(64,151),(65,109),(66,108),(67,107),(68,106),(69,105),(70,112),(71,111),(72,110),(73,182),(74,181),(75,180),(76,179),(77,178),(78,177),(79,184),(80,183),(81,174),(82,173),(83,172),(84,171),(85,170),(86,169),(87,176),(88,175),(89,166),(90,165),(91,164),(92,163),(93,162),(94,161),(95,168),(96,167),(113,190),(114,189),(115,188),(116,187),(117,186),(118,185),(119,192),(120,191),(137,194),(138,193),(139,200),(140,199),(141,198),(142,197),(143,196),(144,195)]])
103 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | ··· | 5L | 8A | 8B | 10A | ··· | 10L | 20A | ··· | 20X | 40A | ··· | 40AV |
| order | 1 | 2 | 2 | 2 | 4 | 5 | ··· | 5 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 100 | 100 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
103 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D5 | D8 | D10 | D20 | D40 |
| kernel | C52⋊5D8 | C5×C40 | C20⋊D5 | C5×C10 | C40 | C52 | C20 | C10 | C5 |
| # reps | 1 | 1 | 2 | 1 | 12 | 2 | 12 | 24 | 48 |
Matrix representation of C52⋊5D8 ►in GL4(𝔽41) generated by
| 0 | 1 | 0 | 0 |
| 40 | 6 | 0 | 0 |
| 0 | 0 | 40 | 6 |
| 0 | 0 | 35 | 35 |
| 0 | 1 | 0 | 0 |
| 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 6 |
| 2 | 13 | 0 | 0 |
| 28 | 39 | 0 | 0 |
| 0 | 0 | 29 | 8 |
| 0 | 0 | 33 | 36 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 39 | 25 |
| 0 | 0 | 13 | 2 |
G:=sub<GL(4,GF(41))| [0,40,0,0,1,6,0,0,0,0,40,35,0,0,6,35],[0,40,0,0,1,6,0,0,0,0,0,40,0,0,1,6],[2,28,0,0,13,39,0,0,0,0,29,33,0,0,8,36],[0,1,0,0,1,0,0,0,0,0,39,13,0,0,25,2] >;
C52⋊5D8 in GAP, Magma, Sage, TeX
C_5^2\rtimes_5D_8
% in TeX
G:=Group("C5^2:5D8"); // GroupNames label
G:=SmallGroup(400,95);
// by ID
G=gap.SmallGroup(400,95);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,79,218,50,1924,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations