metabelian, supersoluble, monomial
Aliases: C52⋊7D8, C20.16D10, D4⋊(C5⋊D5), (C5×D4)⋊1D5, C5⋊3(D4⋊D5), C20⋊D5⋊3C2, C52⋊7C8⋊3C2, (C5×C10).34D4, (D4×C52)⋊2C2, C10.22(C5⋊D4), (C5×C20).12C22, C2.4(C52⋊7D4), C4.1(C2×C5⋊D5), SmallGroup(400,103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊7D8
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 584 in 88 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2, C4, C22, C5, C8, D4, D4, D5, C10, C10, D8, C20, D10, C2×C10, C52, C5⋊2C8, D20, C5×D4, C5⋊D5, C5×C10, C5×C10, D4⋊D5, C5×C20, C2×C5⋊D5, C102, C52⋊7C8, C20⋊D5, D4×C52, C52⋊7D8
Quotients: C1, C2, C22, D4, D5, D8, D10, C5⋊D4, C5⋊D5, D4⋊D5, C2×C5⋊D5, C52⋊7D4, C52⋊7D8
►On 200 points
Generators in S200
(1 44 87 189 15)(2 16 190 88 45)(3 46 81 191 9)(4 10 192 82 47)(5 48 83 185 11)(6 12 186 84 41)(7 42 85 187 13)(8 14 188 86 43)(17 174 168 145 112)(18 105 146 161 175)(19 176 162 147 106)(20 107 148 163 169)(21 170 164 149 108)(22 109 150 165 171)(23 172 166 151 110)(24 111 152 167 173)(25 178 104 139 34)(26 35 140 97 179)(27 180 98 141 36)(28 37 142 99 181)(29 182 100 143 38)(30 39 144 101 183)(31 184 102 137 40)(32 33 138 103 177)(49 64 196 74 117)(50 118 75 197 57)(51 58 198 76 119)(52 120 77 199 59)(53 60 200 78 113)(54 114 79 193 61)(55 62 194 80 115)(56 116 73 195 63)(65 158 95 122 131)(66 132 123 96 159)(67 160 89 124 133)(68 134 125 90 153)(69 154 91 126 135)(70 136 127 92 155)(71 156 93 128 129)(72 130 121 94 157)
(1 161 118 72 29)(2 30 65 119 162)(3 163 120 66 31)(4 32 67 113 164)(5 165 114 68 25)(6 26 69 115 166)(7 167 116 70 27)(8 28 71 117 168)(9 148 52 159 40)(10 33 160 53 149)(11 150 54 153 34)(12 35 154 55 151)(13 152 56 155 36)(14 37 156 49 145)(15 146 50 157 38)(16 39 158 51 147)(17 86 99 128 196)(18 197 121 100 87)(19 88 101 122 198)(20 199 123 102 81)(21 82 103 124 200)(22 193 125 104 83)(23 84 97 126 194)(24 195 127 98 85)(41 179 135 80 172)(42 173 73 136 180)(43 181 129 74 174)(44 175 75 130 182)(45 183 131 76 176)(46 169 77 132 184)(47 177 133 78 170)(48 171 79 134 178)(57 94 143 189 105)(58 106 190 144 95)(59 96 137 191 107)(60 108 192 138 89)(61 90 139 185 109)(62 110 186 140 91)(63 92 141 187 111)(64 112 188 142 93)
(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 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(17 18)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 34)(35 40)(36 39)(37 38)(41 46)(42 45)(43 44)(47 48)(49 50)(51 56)(52 55)(53 54)(57 64)(58 63)(59 62)(60 61)(65 70)(66 69)(67 68)(71 72)(73 76)(74 75)(77 80)(78 79)(81 84)(82 83)(85 88)(86 87)(89 90)(91 96)(92 95)(93 94)(97 102)(98 101)(99 100)(103 104)(105 112)(106 111)(107 110)(108 109)(113 114)(115 120)(116 119)(117 118)(121 128)(122 127)(123 126)(124 125)(129 130)(131 136)(132 135)(133 134)(137 140)(138 139)(141 144)(142 143)(145 146)(147 152)(148 151)(149 150)(153 160)(154 159)(155 158)(156 157)(161 168)(162 167)(163 166)(164 165)(169 172)(170 171)(173 176)(174 175)(177 178)(179 184)(180 183)(181 182)(185 192)(186 191)(187 190)(188 189)(193 200)(194 199)(195 198)(196 197)
G:=sub<Sym(200)| (1,44,87,189,15)(2,16,190,88,45)(3,46,81,191,9)(4,10,192,82,47)(5,48,83,185,11)(6,12,186,84,41)(7,42,85,187,13)(8,14,188,86,43)(17,174,168,145,112)(18,105,146,161,175)(19,176,162,147,106)(20,107,148,163,169)(21,170,164,149,108)(22,109,150,165,171)(23,172,166,151,110)(24,111,152,167,173)(25,178,104,139,34)(26,35,140,97,179)(27,180,98,141,36)(28,37,142,99,181)(29,182,100,143,38)(30,39,144,101,183)(31,184,102,137,40)(32,33,138,103,177)(49,64,196,74,117)(50,118,75,197,57)(51,58,198,76,119)(52,120,77,199,59)(53,60,200,78,113)(54,114,79,193,61)(55,62,194,80,115)(56,116,73,195,63)(65,158,95,122,131)(66,132,123,96,159)(67,160,89,124,133)(68,134,125,90,153)(69,154,91,126,135)(70,136,127,92,155)(71,156,93,128,129)(72,130,121,94,157), (1,161,118,72,29)(2,30,65,119,162)(3,163,120,66,31)(4,32,67,113,164)(5,165,114,68,25)(6,26,69,115,166)(7,167,116,70,27)(8,28,71,117,168)(9,148,52,159,40)(10,33,160,53,149)(11,150,54,153,34)(12,35,154,55,151)(13,152,56,155,36)(14,37,156,49,145)(15,146,50,157,38)(16,39,158,51,147)(17,86,99,128,196)(18,197,121,100,87)(19,88,101,122,198)(20,199,123,102,81)(21,82,103,124,200)(22,193,125,104,83)(23,84,97,126,194)(24,195,127,98,85)(41,179,135,80,172)(42,173,73,136,180)(43,181,129,74,174)(44,175,75,130,182)(45,183,131,76,176)(46,169,77,132,184)(47,177,133,78,170)(48,171,79,134,178)(57,94,143,189,105)(58,106,190,144,95)(59,96,137,191,107)(60,108,192,138,89)(61,90,139,185,109)(62,110,186,140,91)(63,92,141,187,111)(64,112,188,142,93), (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), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(17,18)(19,24)(20,23)(21,22)(25,32)(26,31)(27,30)(28,29)(33,34)(35,40)(36,39)(37,38)(41,46)(42,45)(43,44)(47,48)(49,50)(51,56)(52,55)(53,54)(57,64)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(71,72)(73,76)(74,75)(77,80)(78,79)(81,84)(82,83)(85,88)(86,87)(89,90)(91,96)(92,95)(93,94)(97,102)(98,101)(99,100)(103,104)(105,112)(106,111)(107,110)(108,109)(113,114)(115,120)(116,119)(117,118)(121,128)(122,127)(123,126)(124,125)(129,130)(131,136)(132,135)(133,134)(137,140)(138,139)(141,144)(142,143)(145,146)(147,152)(148,151)(149,150)(153,160)(154,159)(155,158)(156,157)(161,168)(162,167)(163,166)(164,165)(169,172)(170,171)(173,176)(174,175)(177,178)(179,184)(180,183)(181,182)(185,192)(186,191)(187,190)(188,189)(193,200)(194,199)(195,198)(196,197)>;
G:=Group( (1,44,87,189,15)(2,16,190,88,45)(3,46,81,191,9)(4,10,192,82,47)(5,48,83,185,11)(6,12,186,84,41)(7,42,85,187,13)(8,14,188,86,43)(17,174,168,145,112)(18,105,146,161,175)(19,176,162,147,106)(20,107,148,163,169)(21,170,164,149,108)(22,109,150,165,171)(23,172,166,151,110)(24,111,152,167,173)(25,178,104,139,34)(26,35,140,97,179)(27,180,98,141,36)(28,37,142,99,181)(29,182,100,143,38)(30,39,144,101,183)(31,184,102,137,40)(32,33,138,103,177)(49,64,196,74,117)(50,118,75,197,57)(51,58,198,76,119)(52,120,77,199,59)(53,60,200,78,113)(54,114,79,193,61)(55,62,194,80,115)(56,116,73,195,63)(65,158,95,122,131)(66,132,123,96,159)(67,160,89,124,133)(68,134,125,90,153)(69,154,91,126,135)(70,136,127,92,155)(71,156,93,128,129)(72,130,121,94,157), (1,161,118,72,29)(2,30,65,119,162)(3,163,120,66,31)(4,32,67,113,164)(5,165,114,68,25)(6,26,69,115,166)(7,167,116,70,27)(8,28,71,117,168)(9,148,52,159,40)(10,33,160,53,149)(11,150,54,153,34)(12,35,154,55,151)(13,152,56,155,36)(14,37,156,49,145)(15,146,50,157,38)(16,39,158,51,147)(17,86,99,128,196)(18,197,121,100,87)(19,88,101,122,198)(20,199,123,102,81)(21,82,103,124,200)(22,193,125,104,83)(23,84,97,126,194)(24,195,127,98,85)(41,179,135,80,172)(42,173,73,136,180)(43,181,129,74,174)(44,175,75,130,182)(45,183,131,76,176)(46,169,77,132,184)(47,177,133,78,170)(48,171,79,134,178)(57,94,143,189,105)(58,106,190,144,95)(59,96,137,191,107)(60,108,192,138,89)(61,90,139,185,109)(62,110,186,140,91)(63,92,141,187,111)(64,112,188,142,93), (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), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(17,18)(19,24)(20,23)(21,22)(25,32)(26,31)(27,30)(28,29)(33,34)(35,40)(36,39)(37,38)(41,46)(42,45)(43,44)(47,48)(49,50)(51,56)(52,55)(53,54)(57,64)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(71,72)(73,76)(74,75)(77,80)(78,79)(81,84)(82,83)(85,88)(86,87)(89,90)(91,96)(92,95)(93,94)(97,102)(98,101)(99,100)(103,104)(105,112)(106,111)(107,110)(108,109)(113,114)(115,120)(116,119)(117,118)(121,128)(122,127)(123,126)(124,125)(129,130)(131,136)(132,135)(133,134)(137,140)(138,139)(141,144)(142,143)(145,146)(147,152)(148,151)(149,150)(153,160)(154,159)(155,158)(156,157)(161,168)(162,167)(163,166)(164,165)(169,172)(170,171)(173,176)(174,175)(177,178)(179,184)(180,183)(181,182)(185,192)(186,191)(187,190)(188,189)(193,200)(194,199)(195,198)(196,197) );
G=PermutationGroup([[(1,44,87,189,15),(2,16,190,88,45),(3,46,81,191,9),(4,10,192,82,47),(5,48,83,185,11),(6,12,186,84,41),(7,42,85,187,13),(8,14,188,86,43),(17,174,168,145,112),(18,105,146,161,175),(19,176,162,147,106),(20,107,148,163,169),(21,170,164,149,108),(22,109,150,165,171),(23,172,166,151,110),(24,111,152,167,173),(25,178,104,139,34),(26,35,140,97,179),(27,180,98,141,36),(28,37,142,99,181),(29,182,100,143,38),(30,39,144,101,183),(31,184,102,137,40),(32,33,138,103,177),(49,64,196,74,117),(50,118,75,197,57),(51,58,198,76,119),(52,120,77,199,59),(53,60,200,78,113),(54,114,79,193,61),(55,62,194,80,115),(56,116,73,195,63),(65,158,95,122,131),(66,132,123,96,159),(67,160,89,124,133),(68,134,125,90,153),(69,154,91,126,135),(70,136,127,92,155),(71,156,93,128,129),(72,130,121,94,157)], [(1,161,118,72,29),(2,30,65,119,162),(3,163,120,66,31),(4,32,67,113,164),(5,165,114,68,25),(6,26,69,115,166),(7,167,116,70,27),(8,28,71,117,168),(9,148,52,159,40),(10,33,160,53,149),(11,150,54,153,34),(12,35,154,55,151),(13,152,56,155,36),(14,37,156,49,145),(15,146,50,157,38),(16,39,158,51,147),(17,86,99,128,196),(18,197,121,100,87),(19,88,101,122,198),(20,199,123,102,81),(21,82,103,124,200),(22,193,125,104,83),(23,84,97,126,194),(24,195,127,98,85),(41,179,135,80,172),(42,173,73,136,180),(43,181,129,74,174),(44,175,75,130,182),(45,183,131,76,176),(46,169,77,132,184),(47,177,133,78,170),(48,171,79,134,178),(57,94,143,189,105),(58,106,190,144,95),(59,96,137,191,107),(60,108,192,138,89),(61,90,139,185,109),(62,110,186,140,91),(63,92,141,187,111),(64,112,188,142,93)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,16),(14,15),(17,18),(19,24),(20,23),(21,22),(25,32),(26,31),(27,30),(28,29),(33,34),(35,40),(36,39),(37,38),(41,46),(42,45),(43,44),(47,48),(49,50),(51,56),(52,55),(53,54),(57,64),(58,63),(59,62),(60,61),(65,70),(66,69),(67,68),(71,72),(73,76),(74,75),(77,80),(78,79),(81,84),(82,83),(85,88),(86,87),(89,90),(91,96),(92,95),(93,94),(97,102),(98,101),(99,100),(103,104),(105,112),(106,111),(107,110),(108,109),(113,114),(115,120),(116,119),(117,118),(121,128),(122,127),(123,126),(124,125),(129,130),(131,136),(132,135),(133,134),(137,140),(138,139),(141,144),(142,143),(145,146),(147,152),(148,151),(149,150),(153,160),(154,159),(155,158),(156,157),(161,168),(162,167),(163,166),(164,165),(169,172),(170,171),(173,176),(174,175),(177,178),(179,184),(180,183),(181,182),(185,192),(186,191),(187,190),(188,189),(193,200),(194,199),(195,198),(196,197)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | ··· | 5L | 8A | 8B | 10A | ··· | 10L | 10M | ··· | 10AJ | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | 5 | ··· | 5 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 4 | 100 | 2 | 2 | ··· | 2 | 50 | 50 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D5 | D8 | D10 | C5⋊D4 | D4⋊D5 |
| kernel | C52⋊7D8 | C52⋊7C8 | C20⋊D5 | D4×C52 | C5×C10 | C5×D4 | C52 | C20 | C10 | C5 |
| # reps | 1 | 1 | 1 | 1 | 1 | 12 | 2 | 12 | 24 | 12 |
Matrix representation of C52⋊7D8 ►in GL6(𝔽41)
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 16 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 32 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 22 |
| 0 | 0 | 0 | 0 | 13 | 17 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 32 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 22 |
| 0 | 0 | 0 | 0 | 28 | 0 |
G:=sub<GL(6,GF(41))| [10,0,0,0,0,0,0,37,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,18,0,0,0,0,0,0,25,16,0,0,0,0,1,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,25,32,0,0,0,0,1,16,0,0,0,0,0,0,0,13,0,0,0,0,22,17],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,25,32,0,0,0,0,1,16,0,0,0,0,0,0,0,28,0,0,0,0,22,0] >;
C52⋊7D8 in GAP, Magma, Sage, TeX
C_5^2\rtimes_7D_8
% in TeX
G:=Group("C5^2:7D8"); // GroupNames label
G:=SmallGroup(400,103);
// by ID
G=gap.SmallGroup(400,103);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,218,116,50,1924,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations