direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.C23, C20.31C24, D20.28C23, Dic10.27C23, (C2×Q8)⋊27D10, (C22×Q8)⋊4D5, C20.255(C2×D4), (C2×C20).211D4, Q8⋊D5⋊16C22, C4.31(C23×D5), C10⋊4(C8.C22), C5⋊2C8.13C23, (Q8×C10)⋊34C22, C5⋊Q16⋊15C22, (C5×Q8).20C23, Q8.20(C22×D5), (C2×C20).548C23, C4○D20.57C22, C10.150(C22×D4), (C22×C4).274D10, (C22×C10).210D4, C23.93(C5⋊D4), C4.Dic5⋊33C22, (C2×D20).285C22, (C22×C20).280C22, (C2×Dic10).313C22, (Q8×C2×C10)⋊3C2, C5⋊5(C2×C8.C22), (C2×Q8⋊D5)⋊30C2, C4.25(C2×C5⋊D4), (C2×C5⋊Q16)⋊30C2, (C2×C4○D20).24C2, (C2×C10).585(C2×D4), (C2×C4).93(C5⋊D4), (C2×C4.Dic5)⋊27C2, C2.23(C22×C5⋊D4), (C2×C4).240(C22×D5), C22.113(C2×C5⋊D4), (C2×C5⋊2C8).183C22, SmallGroup(320,1480)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20.C23
G = < a,b,c,d,e | a2=b20=c2=1, d2=e2=b10, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b11, cd=dc, ece-1=b5c, ede-1=b10d >
Subgroups: 798 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, D5, C10, C10, C10, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, C20, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C5⋊2C8, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22×C10, C2×C8.C22, C2×C5⋊2C8, C4.Dic5, Q8⋊D5, C5⋊Q16, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C2×C5⋊D4, C22×C20, C22×C20, Q8×C10, Q8×C10, C2×C4.Dic5, C2×Q8⋊D5, C20.C23, C2×C5⋊Q16, C2×C4○D20, Q8×C2×C10, C2×C20.C23
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C8.C22, C22×D4, C5⋊D4, C22×D5, C2×C8.C22, C2×C5⋊D4, C23×D5, C20.C23, C22×C5⋊D4, C2×C20.C23
►On 160 points
Generators in S160
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 118)(22 119)(23 120)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 141)
(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)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(98 100)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)(134 140)(135 139)(136 138)(141 146)(142 145)(143 144)(147 160)(148 159)(149 158)(150 157)(151 156)(152 155)(153 154)
(1 137 11 127)(2 138 12 128)(3 139 13 129)(4 140 14 130)(5 121 15 131)(6 122 16 132)(7 123 17 133)(8 124 18 134)(9 125 19 135)(10 126 20 136)(21 156 31 146)(22 157 32 147)(23 158 33 148)(24 159 34 149)(25 160 35 150)(26 141 36 151)(27 142 37 152)(28 143 38 153)(29 144 39 154)(30 145 40 155)(41 83 51 93)(42 84 52 94)(43 85 53 95)(44 86 54 96)(45 87 55 97)(46 88 56 98)(47 89 57 99)(48 90 58 100)(49 91 59 81)(50 92 60 82)(61 114 71 104)(62 115 72 105)(63 116 73 106)(64 117 74 107)(65 118 75 108)(66 119 76 109)(67 120 77 110)(68 101 78 111)(69 102 79 112)(70 103 80 113)
(1 26 11 36)(2 37 12 27)(3 28 13 38)(4 39 14 29)(5 30 15 40)(6 21 16 31)(7 32 17 22)(8 23 18 33)(9 34 19 24)(10 25 20 35)(41 74 51 64)(42 65 52 75)(43 76 53 66)(44 67 54 77)(45 78 55 68)(46 69 56 79)(47 80 57 70)(48 71 58 61)(49 62 59 72)(50 73 60 63)(81 115 91 105)(82 106 92 116)(83 117 93 107)(84 108 94 118)(85 119 95 109)(86 110 96 120)(87 101 97 111)(88 112 98 102)(89 103 99 113)(90 114 100 104)(121 155 131 145)(122 146 132 156)(123 157 133 147)(124 148 134 158)(125 159 135 149)(126 150 136 160)(127 141 137 151)(128 152 138 142)(129 143 139 153)(130 154 140 144)
G:=sub<Sym(160)| (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,118)(22,119)(23,120)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,141), (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), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(134,140)(135,139)(136,138)(141,146)(142,145)(143,144)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154), (1,137,11,127)(2,138,12,128)(3,139,13,129)(4,140,14,130)(5,121,15,131)(6,122,16,132)(7,123,17,133)(8,124,18,134)(9,125,19,135)(10,126,20,136)(21,156,31,146)(22,157,32,147)(23,158,33,148)(24,159,34,149)(25,160,35,150)(26,141,36,151)(27,142,37,152)(28,143,38,153)(29,144,39,154)(30,145,40,155)(41,83,51,93)(42,84,52,94)(43,85,53,95)(44,86,54,96)(45,87,55,97)(46,88,56,98)(47,89,57,99)(48,90,58,100)(49,91,59,81)(50,92,60,82)(61,114,71,104)(62,115,72,105)(63,116,73,106)(64,117,74,107)(65,118,75,108)(66,119,76,109)(67,120,77,110)(68,101,78,111)(69,102,79,112)(70,103,80,113), (1,26,11,36)(2,37,12,27)(3,28,13,38)(4,39,14,29)(5,30,15,40)(6,21,16,31)(7,32,17,22)(8,23,18,33)(9,34,19,24)(10,25,20,35)(41,74,51,64)(42,65,52,75)(43,76,53,66)(44,67,54,77)(45,78,55,68)(46,69,56,79)(47,80,57,70)(48,71,58,61)(49,62,59,72)(50,73,60,63)(81,115,91,105)(82,106,92,116)(83,117,93,107)(84,108,94,118)(85,119,95,109)(86,110,96,120)(87,101,97,111)(88,112,98,102)(89,103,99,113)(90,114,100,104)(121,155,131,145)(122,146,132,156)(123,157,133,147)(124,148,134,158)(125,159,135,149)(126,150,136,160)(127,141,137,151)(128,152,138,142)(129,143,139,153)(130,154,140,144)>;
G:=Group( (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,118)(22,119)(23,120)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,141), (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), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(134,140)(135,139)(136,138)(141,146)(142,145)(143,144)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154), (1,137,11,127)(2,138,12,128)(3,139,13,129)(4,140,14,130)(5,121,15,131)(6,122,16,132)(7,123,17,133)(8,124,18,134)(9,125,19,135)(10,126,20,136)(21,156,31,146)(22,157,32,147)(23,158,33,148)(24,159,34,149)(25,160,35,150)(26,141,36,151)(27,142,37,152)(28,143,38,153)(29,144,39,154)(30,145,40,155)(41,83,51,93)(42,84,52,94)(43,85,53,95)(44,86,54,96)(45,87,55,97)(46,88,56,98)(47,89,57,99)(48,90,58,100)(49,91,59,81)(50,92,60,82)(61,114,71,104)(62,115,72,105)(63,116,73,106)(64,117,74,107)(65,118,75,108)(66,119,76,109)(67,120,77,110)(68,101,78,111)(69,102,79,112)(70,103,80,113), (1,26,11,36)(2,37,12,27)(3,28,13,38)(4,39,14,29)(5,30,15,40)(6,21,16,31)(7,32,17,22)(8,23,18,33)(9,34,19,24)(10,25,20,35)(41,74,51,64)(42,65,52,75)(43,76,53,66)(44,67,54,77)(45,78,55,68)(46,69,56,79)(47,80,57,70)(48,71,58,61)(49,62,59,72)(50,73,60,63)(81,115,91,105)(82,106,92,116)(83,117,93,107)(84,108,94,118)(85,119,95,109)(86,110,96,120)(87,101,97,111)(88,112,98,102)(89,103,99,113)(90,114,100,104)(121,155,131,145)(122,146,132,156)(123,157,133,147)(124,148,134,158)(125,159,135,149)(126,150,136,160)(127,141,137,151)(128,152,138,142)(129,143,139,153)(130,154,140,144) );
G=PermutationGroup([[(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,100),(13,81),(14,82),(15,83),(16,84),(17,85),(18,86),(19,87),(20,88),(21,118),(22,119),(23,120),(24,101),(25,102),(26,103),(27,104),(28,105),(29,106),(30,107),(31,108),(32,109),(33,110),(34,111),(35,112),(36,113),(37,114),(38,115),(39,116),(40,117),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,142),(62,143),(63,144),(64,145),(65,146),(66,147),(67,148),(68,149),(69,150),(70,151),(71,152),(72,153),(73,154),(74,155),(75,156),(76,157),(77,158),(78,159),(79,160),(80,141)], [(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)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(37,40),(38,39),(41,53),(42,52),(43,51),(44,50),(45,49),(46,48),(54,60),(55,59),(56,58),(61,64),(62,63),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(98,100),(101,110),(102,109),(103,108),(104,107),(105,106),(111,120),(112,119),(113,118),(114,117),(115,116),(121,133),(122,132),(123,131),(124,130),(125,129),(126,128),(134,140),(135,139),(136,138),(141,146),(142,145),(143,144),(147,160),(148,159),(149,158),(150,157),(151,156),(152,155),(153,154)], [(1,137,11,127),(2,138,12,128),(3,139,13,129),(4,140,14,130),(5,121,15,131),(6,122,16,132),(7,123,17,133),(8,124,18,134),(9,125,19,135),(10,126,20,136),(21,156,31,146),(22,157,32,147),(23,158,33,148),(24,159,34,149),(25,160,35,150),(26,141,36,151),(27,142,37,152),(28,143,38,153),(29,144,39,154),(30,145,40,155),(41,83,51,93),(42,84,52,94),(43,85,53,95),(44,86,54,96),(45,87,55,97),(46,88,56,98),(47,89,57,99),(48,90,58,100),(49,91,59,81),(50,92,60,82),(61,114,71,104),(62,115,72,105),(63,116,73,106),(64,117,74,107),(65,118,75,108),(66,119,76,109),(67,120,77,110),(68,101,78,111),(69,102,79,112),(70,103,80,113)], [(1,26,11,36),(2,37,12,27),(3,28,13,38),(4,39,14,29),(5,30,15,40),(6,21,16,31),(7,32,17,22),(8,23,18,33),(9,34,19,24),(10,25,20,35),(41,74,51,64),(42,65,52,75),(43,76,53,66),(44,67,54,77),(45,78,55,68),(46,69,56,79),(47,80,57,70),(48,71,58,61),(49,62,59,72),(50,73,60,63),(81,115,91,105),(82,106,92,116),(83,117,93,107),(84,108,94,118),(85,119,95,109),(86,110,96,120),(87,101,97,111),(88,112,98,102),(89,103,99,113),(90,114,100,104),(121,155,131,145),(122,146,132,156),(123,157,133,147),(124,148,134,158),(125,159,135,149),(126,150,136,160),(127,141,137,151),(128,152,138,142),(129,143,139,153),(130,154,140,144)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8.C22 | C20.C23 |
| kernel | C2×C20.C23 | C2×C4.Dic5 | C2×Q8⋊D5 | C20.C23 | C2×C5⋊Q16 | C2×C4○D20 | Q8×C2×C10 | C2×C20 | C22×C10 | C22×Q8 | C22×C4 | C2×Q8 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 1 | 1 | 3 | 1 | 2 | 2 | 12 | 12 | 4 | 2 | 8 |
Matrix representation of C2×C20.C23 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 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 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 35 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 6 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 6 |
| 0 | 0 | 0 | 0 | 35 | 23 |
| 0 | 0 | 23 | 35 | 0 | 0 |
| 0 | 0 | 6 | 18 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 36 | 20 | 5 |
| 0 | 0 | 5 | 20 | 36 | 31 |
| 0 | 0 | 20 | 5 | 10 | 5 |
| 0 | 0 | 36 | 31 | 36 | 21 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,35,1,0,0,0,0,40,0,0,0,6,40,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,0,0,0,0,35,1,0,0,0,0,6,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,23,6,0,0,0,0,35,18,0,0,18,35,0,0,0,0,6,23,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,31,5,20,36,0,0,36,20,5,31,0,0,20,36,10,36,0,0,5,31,5,21] >;
C2×C20.C23 in GAP, Magma, Sage, TeX
C_2\times C_{20}.C_2^3 % in TeX
G:=Group("C2xC20.C2^3"); // GroupNames label
G:=SmallGroup(320,1480);
// by ID
G=gap.SmallGroup(320,1480);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,136,1684,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^20=c^2=1,d^2=e^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^11,c*d=d*c,e*c*e^-1=b^5*c,e*d*e^-1=b^10*d>;
// generators/relations