metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23⋊2D20, C24.15D10, (C2×C20)⋊6D4, (C2×Dic5)⋊5D4, (C22×D5)⋊4D4, (C22×C10)⋊7D4, (C22×D20)⋊3C2, C5⋊2(C23⋊2D4), C10.33C22≀C2, C2.6(C20⋊D4), C2.7(C20⋊7D4), (C22×C4).35D10, C22.242(D4×D5), C2.8(C23⋊D10), C10.59(C4⋊D4), C10.13(C4⋊1D4), C22.126(C2×D20), C2.34(C22⋊D20), C2.34(D10⋊D4), (C23×C10).43C22, (C22×C20).61C22, (C23×D5).16C22, C23.372(C22×D5), C10.10C42⋊32C2, C22.100(C4○D20), (C22×C10).334C23, (C22×Dic5).46C22, (C2×C4)⋊3(C5⋊D4), (C2×C22⋊C4)⋊8D5, (C22×C5⋊D4)⋊1C2, (C2×D10⋊C4)⋊8C2, (C10×C22⋊C4)⋊11C2, (C2×C10).325(C2×D4), (C2×C10).80(C4○D4), C22.128(C2×C5⋊D4), SmallGroup(320,587)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊2D20
G = < a,b,c,d,e | a2=b2=c2=d20=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1526 in 322 conjugacy classes, 67 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C22×D4, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C23⋊2D4, D10⋊C4, C5×C22⋊C4, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C23×D5, C23×C10, C10.10C42, C2×D10⋊C4, C10×C22⋊C4, C22×D20, C22×C5⋊D4, C23⋊2D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22≀C2, C4⋊D4, C4⋊1D4, D20, C5⋊D4, C22×D5, C23⋊2D4, C2×D20, C4○D20, D4×D5, C2×C5⋊D4, C22⋊D20, D10⋊D4, C20⋊7D4, C23⋊D10, C20⋊D4, C23⋊2D20
►On 160 points
Generators in S160
(1 29)(2 90)(3 31)(4 92)(5 33)(6 94)(7 35)(8 96)(9 37)(10 98)(11 39)(12 100)(13 21)(14 82)(15 23)(16 84)(17 25)(18 86)(19 27)(20 88)(22 146)(24 148)(26 150)(28 152)(30 154)(32 156)(34 158)(36 160)(38 142)(40 144)(41 128)(42 72)(43 130)(44 74)(45 132)(46 76)(47 134)(48 78)(49 136)(50 80)(51 138)(52 62)(53 140)(54 64)(55 122)(56 66)(57 124)(58 68)(59 126)(60 70)(61 111)(63 113)(65 115)(67 117)(69 119)(71 101)(73 103)(75 105)(77 107)(79 109)(81 145)(83 147)(85 149)(87 151)(89 153)(91 155)(93 157)(95 159)(97 141)(99 143)(102 129)(104 131)(106 133)(108 135)(110 137)(112 139)(114 121)(116 123)(118 125)(120 127)
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 113)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 101)(37 102)(38 103)(39 104)(40 105)(41 96)(42 97)(43 98)(44 99)(45 100)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)(61 150)(62 151)(63 152)(64 153)(65 154)(66 155)(67 156)(68 157)(69 158)(70 159)(71 160)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 148)(80 149)
(1 153)(2 154)(3 155)(4 156)(5 157)(6 158)(7 159)(8 160)(9 141)(10 142)(11 143)(12 144)(13 145)(14 146)(15 147)(16 148)(17 149)(18 150)(19 151)(20 152)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 138)(62 139)(63 140)(64 121)(65 122)(66 123)(67 124)(68 125)(69 126)(70 127)(71 128)(72 129)(73 130)(74 131)(75 132)(76 133)(77 134)(78 135)(79 136)(80 137)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 101)(22 120)(23 119)(24 118)(25 117)(26 116)(27 115)(28 114)(29 113)(30 112)(31 111)(32 110)(33 109)(34 108)(35 107)(36 106)(37 105)(38 104)(39 103)(40 102)(41 81)(42 100)(43 99)(44 98)(45 97)(46 96)(47 95)(48 94)(49 93)(50 92)(51 91)(52 90)(53 89)(54 88)(55 87)(56 86)(57 85)(58 84)(59 83)(60 82)(61 66)(62 65)(63 64)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)(141 144)(142 143)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)(151 154)(152 153)
G:=sub<Sym(160)| (1,29)(2,90)(3,31)(4,92)(5,33)(6,94)(7,35)(8,96)(9,37)(10,98)(11,39)(12,100)(13,21)(14,82)(15,23)(16,84)(17,25)(18,86)(19,27)(20,88)(22,146)(24,148)(26,150)(28,152)(30,154)(32,156)(34,158)(36,160)(38,142)(40,144)(41,128)(42,72)(43,130)(44,74)(45,132)(46,76)(47,134)(48,78)(49,136)(50,80)(51,138)(52,62)(53,140)(54,64)(55,122)(56,66)(57,124)(58,68)(59,126)(60,70)(61,111)(63,113)(65,115)(67,117)(69,119)(71,101)(73,103)(75,105)(77,107)(79,109)(81,145)(83,147)(85,149)(87,151)(89,153)(91,155)(93,157)(95,159)(97,141)(99,143)(102,129)(104,131)(106,133)(108,135)(110,137)(112,139)(114,121)(116,123)(118,125)(120,127), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,101)(37,102)(38,103)(39,104)(40,105)(41,96)(42,97)(43,98)(44,99)(45,100)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149), (1,153)(2,154)(3,155)(4,156)(5,157)(6,158)(7,159)(8,160)(9,141)(10,142)(11,143)(12,144)(13,145)(14,146)(15,147)(16,148)(17,149)(18,150)(19,151)(20,152)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,138)(62,139)(63,140)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133)(77,134)(78,135)(79,136)(80,137), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,101)(22,120)(23,119)(24,118)(25,117)(26,116)(27,115)(28,114)(29,113)(30,112)(31,111)(32,110)(33,109)(34,108)(35,107)(36,106)(37,105)(38,104)(39,103)(40,102)(41,81)(42,100)(43,99)(44,98)(45,97)(46,96)(47,95)(48,94)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(57,85)(58,84)(59,83)(60,82)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(141,144)(142,143)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,154)(152,153)>;
G:=Group( (1,29)(2,90)(3,31)(4,92)(5,33)(6,94)(7,35)(8,96)(9,37)(10,98)(11,39)(12,100)(13,21)(14,82)(15,23)(16,84)(17,25)(18,86)(19,27)(20,88)(22,146)(24,148)(26,150)(28,152)(30,154)(32,156)(34,158)(36,160)(38,142)(40,144)(41,128)(42,72)(43,130)(44,74)(45,132)(46,76)(47,134)(48,78)(49,136)(50,80)(51,138)(52,62)(53,140)(54,64)(55,122)(56,66)(57,124)(58,68)(59,126)(60,70)(61,111)(63,113)(65,115)(67,117)(69,119)(71,101)(73,103)(75,105)(77,107)(79,109)(81,145)(83,147)(85,149)(87,151)(89,153)(91,155)(93,157)(95,159)(97,141)(99,143)(102,129)(104,131)(106,133)(108,135)(110,137)(112,139)(114,121)(116,123)(118,125)(120,127), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,101)(37,102)(38,103)(39,104)(40,105)(41,96)(42,97)(43,98)(44,99)(45,100)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149), (1,153)(2,154)(3,155)(4,156)(5,157)(6,158)(7,159)(8,160)(9,141)(10,142)(11,143)(12,144)(13,145)(14,146)(15,147)(16,148)(17,149)(18,150)(19,151)(20,152)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,138)(62,139)(63,140)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133)(77,134)(78,135)(79,136)(80,137), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,101)(22,120)(23,119)(24,118)(25,117)(26,116)(27,115)(28,114)(29,113)(30,112)(31,111)(32,110)(33,109)(34,108)(35,107)(36,106)(37,105)(38,104)(39,103)(40,102)(41,81)(42,100)(43,99)(44,98)(45,97)(46,96)(47,95)(48,94)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(57,85)(58,84)(59,83)(60,82)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(141,144)(142,143)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,154)(152,153) );
G=PermutationGroup([[(1,29),(2,90),(3,31),(4,92),(5,33),(6,94),(7,35),(8,96),(9,37),(10,98),(11,39),(12,100),(13,21),(14,82),(15,23),(16,84),(17,25),(18,86),(19,27),(20,88),(22,146),(24,148),(26,150),(28,152),(30,154),(32,156),(34,158),(36,160),(38,142),(40,144),(41,128),(42,72),(43,130),(44,74),(45,132),(46,76),(47,134),(48,78),(49,136),(50,80),(51,138),(52,62),(53,140),(54,64),(55,122),(56,66),(57,124),(58,68),(59,126),(60,70),(61,111),(63,113),(65,115),(67,117),(69,119),(71,101),(73,103),(75,105),(77,107),(79,109),(81,145),(83,147),(85,149),(87,151),(89,153),(91,155),(93,157),(95,159),(97,141),(99,143),(102,129),(104,131),(106,133),(108,135),(110,137),(112,139),(114,121),(116,123),(118,125),(120,127)], [(1,121),(2,122),(3,123),(4,124),(5,125),(6,126),(7,127),(8,128),(9,129),(10,130),(11,131),(12,132),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,106),(22,107),(23,108),(24,109),(25,110),(26,111),(27,112),(28,113),(29,114),(30,115),(31,116),(32,117),(33,118),(34,119),(35,120),(36,101),(37,102),(38,103),(39,104),(40,105),(41,96),(42,97),(43,98),(44,99),(45,100),(46,81),(47,82),(48,83),(49,84),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91),(57,92),(58,93),(59,94),(60,95),(61,150),(62,151),(63,152),(64,153),(65,154),(66,155),(67,156),(68,157),(69,158),(70,159),(71,160),(72,141),(73,142),(74,143),(75,144),(76,145),(77,146),(78,147),(79,148),(80,149)], [(1,153),(2,154),(3,155),(4,156),(5,157),(6,158),(7,159),(8,160),(9,141),(10,142),(11,143),(12,144),(13,145),(14,146),(15,147),(16,148),(17,149),(18,150),(19,151),(20,152),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(61,138),(62,139),(63,140),(64,121),(65,122),(66,123),(67,124),(68,125),(69,126),(70,127),(71,128),(72,129),(73,130),(74,131),(75,132),(76,133),(77,134),(78,135),(79,136),(80,137)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,101),(22,120),(23,119),(24,118),(25,117),(26,116),(27,115),(28,114),(29,113),(30,112),(31,111),(32,110),(33,109),(34,108),(35,107),(36,106),(37,105),(38,104),(39,103),(40,102),(41,81),(42,100),(43,99),(44,98),(45,97),(46,96),(47,95),(48,94),(49,93),(50,92),(51,91),(52,90),(53,89),(54,88),(55,87),(56,86),(57,85),(58,84),(59,83),(60,82),(61,66),(62,65),(63,64),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(121,140),(122,139),(123,138),(124,137),(125,136),(126,135),(127,134),(128,133),(129,132),(130,131),(141,144),(142,143),(145,160),(146,159),(147,158),(148,157),(149,156),(150,155),(151,154),(152,153)]])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D20 | C4○D20 | D4×D5 |
| kernel | C23⋊2D20 | C10.10C42 | C2×D10⋊C4 | C10×C22⋊C4 | C22×D20 | C22×C5⋊D4 | C2×Dic5 | C2×C20 | C22×D5 | C22×C10 | C2×C22⋊C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 8 | 8 | 8 | 8 |
Matrix representation of C23⋊2D20 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 1 |
| 0 | 0 | 0 | 0 | 40 | 17 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 30 | 14 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 30 | 32 |
G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,39,1,0,0,0,0,0,0,24,40,0,0,0,0,1,17],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,2,40,0,0,0,0,0,0,9,30,0,0,0,0,11,14],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,40,0,0,0,0,0,0,9,30,0,0,0,0,11,32] >;
C23⋊2D20 in GAP, Magma, Sage, TeX
C_2^3\rtimes_2D_{20} % in TeX
G:=Group("C2^3:2D20"); // GroupNames label
G:=SmallGroup(320,587);
// by ID
G=gap.SmallGroup(320,587);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=e^2=1,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations