metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.45D20, C24.49D10, C10.74(C4×D4), (C2×Dic5)⋊17D4, C23.26(C4×D5), C10.32C22≀C2, (C23×Dic5)⋊1C2, (C22×C4).33D10, C22.103(D4×D5), (C22×C10).68D4, C22.44(C2×D20), C2.5(C22⋊D20), C10.86(C4⋊D4), C23.36(C5⋊D4), C5⋊4(C23.23D4), C22⋊1(D10⋊C4), C2.4(Dic5⋊D4), (C22×C20).26C22, (C23×C10).41C22, (C23×D5).14C22, C23.285(C22×D5), C10.10C42⋊16C2, C2.29(Dic5⋊4D4), C22.50(D4⋊2D5), (C22×C10).332C23, C2.5(C22.D20), C10.33(C22.D4), (C22×Dic5).209C22, (C2×C5⋊D4)⋊11C4, (C2×C22⋊C4)⋊5D5, (C10×C22⋊C4)⋊3C2, (C2×Dic5)⋊8(C2×C4), (C22×D5)⋊6(C2×C4), (C2×D10⋊C4)⋊6C2, C22.129(C2×C4×D5), (C2×C10)⋊5(C22⋊C4), (C2×C10).324(C2×D4), C10.78(C2×C22⋊C4), (C22×C5⋊D4).4C2, C22.53(C2×C5⋊D4), C2.10(C2×D10⋊C4), (C2×C10).147(C4○D4), (C2×C10).212(C22×C4), (C22×C10).123(C2×C4), SmallGroup(320,585)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.45D20
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=b, f2=cb=bc, ab=ba, ac=ca, eae-1=ad=da, af=fa, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >
Subgroups: 1118 in 286 conjugacy classes, 83 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×8], C22 [×3], C22 [×8], C22 [×22], C5, C2×C4 [×26], D4 [×8], C23, C23 [×6], C23 [×12], D5 [×2], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×6], C22×C4 [×2], C22×C4 [×9], C2×D4 [×8], C24, C24, Dic5 [×6], C20 [×2], D10 [×10], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2.C42 [×2], C2×C22⋊C4, C2×C22⋊C4 [×2], C23×C4, C22×D4, C2×Dic5 [×6], C2×Dic5 [×14], C5⋊D4 [×8], C2×C20 [×6], C22×D5 [×2], C22×D5 [×6], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.23D4, D10⋊C4 [×4], C5×C22⋊C4 [×2], C22×Dic5, C22×Dic5 [×2], C22×Dic5 [×6], C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], C23×D5, C23×C10, C10.10C42 [×2], C2×D10⋊C4 [×2], C10×C22⋊C4, C23×Dic5, C22×C5⋊D4, C23.45D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], D10 [×3], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.23D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, D4×D5 [×2], D4⋊2D5 [×2], C2×C5⋊D4, Dic5⋊4D4 [×2], C22⋊D20, C22.D20, C2×D10⋊C4, Dic5⋊D4 [×2], C23.45D20
►On 160 points
Generators in S160
(1 25)(2 76)(3 27)(4 78)(5 29)(6 80)(7 31)(8 62)(9 33)(10 64)(11 35)(12 66)(13 37)(14 68)(15 39)(16 70)(17 21)(18 72)(19 23)(20 74)(22 58)(24 60)(26 42)(28 44)(30 46)(32 48)(34 50)(36 52)(38 54)(40 56)(41 75)(43 77)(45 79)(47 61)(49 63)(51 65)(53 67)(55 69)(57 71)(59 73)(81 133)(82 109)(83 135)(84 111)(85 137)(86 113)(87 139)(88 115)(89 121)(90 117)(91 123)(92 119)(93 125)(94 101)(95 127)(96 103)(97 129)(98 105)(99 131)(100 107)(102 146)(104 148)(106 150)(108 152)(110 154)(112 156)(114 158)(116 160)(118 142)(120 144)(122 141)(124 143)(126 145)(128 147)(130 149)(132 151)(134 153)(136 155)(138 157)(140 159)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 151)(2 152)(3 153)(4 154)(5 155)(6 156)(7 157)(8 158)(9 159)(10 160)(11 141)(12 142)(13 143)(14 144)(15 145)(16 146)(17 147)(18 148)(19 149)(20 150)(21 128)(22 129)(23 130)(24 131)(25 132)(26 133)(27 134)(28 135)(29 136)(30 137)(31 138)(32 139)(33 140)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 100)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 101)(70 102)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(81 152)(82 153)(83 154)(84 155)(85 156)(86 157)(87 158)(88 159)(89 160)(90 141)(91 142)(92 143)(93 144)(94 145)(95 146)(96 147)(97 148)(98 149)(99 150)(100 151)(101 126)(102 127)(103 128)(104 129)(105 130)(106 131)(107 132)(108 133)(109 134)(110 135)(111 136)(112 137)(113 138)(114 139)(115 140)(116 121)(117 122)(118 123)(119 124)(120 125)
(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 10 141 150)(2 149 142 9)(3 8 143 148)(4 147 144 7)(5 6 145 146)(11 20 151 160)(12 159 152 19)(13 18 153 158)(14 157 154 17)(15 16 155 156)(21 68 138 110)(22 109 139 67)(23 66 140 108)(24 107 121 65)(25 64 122 106)(26 105 123 63)(27 62 124 104)(28 103 125 61)(29 80 126 102)(30 101 127 79)(31 78 128 120)(32 119 129 77)(33 76 130 118)(34 117 131 75)(35 74 132 116)(36 115 133 73)(37 72 134 114)(38 113 135 71)(39 70 136 112)(40 111 137 69)(41 50 90 99)(42 98 91 49)(43 48 92 97)(44 96 93 47)(45 46 94 95)(51 60 100 89)(52 88 81 59)(53 58 82 87)(54 86 83 57)(55 56 84 85)
G:=sub<Sym(160)| (1,25)(2,76)(3,27)(4,78)(5,29)(6,80)(7,31)(8,62)(9,33)(10,64)(11,35)(12,66)(13,37)(14,68)(15,39)(16,70)(17,21)(18,72)(19,23)(20,74)(22,58)(24,60)(26,42)(28,44)(30,46)(32,48)(34,50)(36,52)(38,54)(40,56)(41,75)(43,77)(45,79)(47,61)(49,63)(51,65)(53,67)(55,69)(57,71)(59,73)(81,133)(82,109)(83,135)(84,111)(85,137)(86,113)(87,139)(88,115)(89,121)(90,117)(91,123)(92,119)(93,125)(94,101)(95,127)(96,103)(97,129)(98,105)(99,131)(100,107)(102,146)(104,148)(106,150)(108,152)(110,154)(112,156)(114,158)(116,160)(118,142)(120,144)(122,141)(124,143)(126,145)(128,147)(130,149)(132,151)(134,153)(136,155)(138,157)(140,159), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,151)(2,152)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,159)(10,160)(11,141)(12,142)(13,143)(14,144)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,128)(22,129)(23,130)(24,131)(25,132)(26,133)(27,134)(28,135)(29,136)(30,137)(31,138)(32,139)(33,140)(34,121)(35,122)(36,123)(37,124)(38,125)(39,126)(40,127)(41,100)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,141)(91,142)(92,143)(93,144)(94,145)(95,146)(96,147)(97,148)(98,149)(99,150)(100,151)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,121)(117,122)(118,123)(119,124)(120,125), (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), (1,10,141,150)(2,149,142,9)(3,8,143,148)(4,147,144,7)(5,6,145,146)(11,20,151,160)(12,159,152,19)(13,18,153,158)(14,157,154,17)(15,16,155,156)(21,68,138,110)(22,109,139,67)(23,66,140,108)(24,107,121,65)(25,64,122,106)(26,105,123,63)(27,62,124,104)(28,103,125,61)(29,80,126,102)(30,101,127,79)(31,78,128,120)(32,119,129,77)(33,76,130,118)(34,117,131,75)(35,74,132,116)(36,115,133,73)(37,72,134,114)(38,113,135,71)(39,70,136,112)(40,111,137,69)(41,50,90,99)(42,98,91,49)(43,48,92,97)(44,96,93,47)(45,46,94,95)(51,60,100,89)(52,88,81,59)(53,58,82,87)(54,86,83,57)(55,56,84,85)>;
G:=Group( (1,25)(2,76)(3,27)(4,78)(5,29)(6,80)(7,31)(8,62)(9,33)(10,64)(11,35)(12,66)(13,37)(14,68)(15,39)(16,70)(17,21)(18,72)(19,23)(20,74)(22,58)(24,60)(26,42)(28,44)(30,46)(32,48)(34,50)(36,52)(38,54)(40,56)(41,75)(43,77)(45,79)(47,61)(49,63)(51,65)(53,67)(55,69)(57,71)(59,73)(81,133)(82,109)(83,135)(84,111)(85,137)(86,113)(87,139)(88,115)(89,121)(90,117)(91,123)(92,119)(93,125)(94,101)(95,127)(96,103)(97,129)(98,105)(99,131)(100,107)(102,146)(104,148)(106,150)(108,152)(110,154)(112,156)(114,158)(116,160)(118,142)(120,144)(122,141)(124,143)(126,145)(128,147)(130,149)(132,151)(134,153)(136,155)(138,157)(140,159), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,151)(2,152)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,159)(10,160)(11,141)(12,142)(13,143)(14,144)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,128)(22,129)(23,130)(24,131)(25,132)(26,133)(27,134)(28,135)(29,136)(30,137)(31,138)(32,139)(33,140)(34,121)(35,122)(36,123)(37,124)(38,125)(39,126)(40,127)(41,100)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,141)(91,142)(92,143)(93,144)(94,145)(95,146)(96,147)(97,148)(98,149)(99,150)(100,151)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,121)(117,122)(118,123)(119,124)(120,125), (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), (1,10,141,150)(2,149,142,9)(3,8,143,148)(4,147,144,7)(5,6,145,146)(11,20,151,160)(12,159,152,19)(13,18,153,158)(14,157,154,17)(15,16,155,156)(21,68,138,110)(22,109,139,67)(23,66,140,108)(24,107,121,65)(25,64,122,106)(26,105,123,63)(27,62,124,104)(28,103,125,61)(29,80,126,102)(30,101,127,79)(31,78,128,120)(32,119,129,77)(33,76,130,118)(34,117,131,75)(35,74,132,116)(36,115,133,73)(37,72,134,114)(38,113,135,71)(39,70,136,112)(40,111,137,69)(41,50,90,99)(42,98,91,49)(43,48,92,97)(44,96,93,47)(45,46,94,95)(51,60,100,89)(52,88,81,59)(53,58,82,87)(54,86,83,57)(55,56,84,85) );
G=PermutationGroup([(1,25),(2,76),(3,27),(4,78),(5,29),(6,80),(7,31),(8,62),(9,33),(10,64),(11,35),(12,66),(13,37),(14,68),(15,39),(16,70),(17,21),(18,72),(19,23),(20,74),(22,58),(24,60),(26,42),(28,44),(30,46),(32,48),(34,50),(36,52),(38,54),(40,56),(41,75),(43,77),(45,79),(47,61),(49,63),(51,65),(53,67),(55,69),(57,71),(59,73),(81,133),(82,109),(83,135),(84,111),(85,137),(86,113),(87,139),(88,115),(89,121),(90,117),(91,123),(92,119),(93,125),(94,101),(95,127),(96,103),(97,129),(98,105),(99,131),(100,107),(102,146),(104,148),(106,150),(108,152),(110,154),(112,156),(114,158),(116,160),(118,142),(120,144),(122,141),(124,143),(126,145),(128,147),(130,149),(132,151),(134,153),(136,155),(138,157),(140,159)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160)], [(1,151),(2,152),(3,153),(4,154),(5,155),(6,156),(7,157),(8,158),(9,159),(10,160),(11,141),(12,142),(13,143),(14,144),(15,145),(16,146),(17,147),(18,148),(19,149),(20,150),(21,128),(22,129),(23,130),(24,131),(25,132),(26,133),(27,134),(28,135),(29,136),(30,137),(31,138),(32,139),(33,140),(34,121),(35,122),(36,123),(37,124),(38,125),(39,126),(40,127),(41,100),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,88),(50,89),(51,90),(52,91),(53,92),(54,93),(55,94),(56,95),(57,96),(58,97),(59,98),(60,99),(61,113),(62,114),(63,115),(64,116),(65,117),(66,118),(67,119),(68,120),(69,101),(70,102),(71,103),(72,104),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(81,152),(82,153),(83,154),(84,155),(85,156),(86,157),(87,158),(88,159),(89,160),(90,141),(91,142),(92,143),(93,144),(94,145),(95,146),(96,147),(97,148),(98,149),(99,150),(100,151),(101,126),(102,127),(103,128),(104,129),(105,130),(106,131),(107,132),(108,133),(109,134),(110,135),(111,136),(112,137),(113,138),(114,139),(115,140),(116,121),(117,122),(118,123),(119,124),(120,125)], [(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)], [(1,10,141,150),(2,149,142,9),(3,8,143,148),(4,147,144,7),(5,6,145,146),(11,20,151,160),(12,159,152,19),(13,18,153,158),(14,157,154,17),(15,16,155,156),(21,68,138,110),(22,109,139,67),(23,66,140,108),(24,107,121,65),(25,64,122,106),(26,105,123,63),(27,62,124,104),(28,103,125,61),(29,80,126,102),(30,101,127,79),(31,78,128,120),(32,119,129,77),(33,76,130,118),(34,117,131,75),(35,74,132,116),(36,115,133,73),(37,72,134,114),(38,113,135,71),(39,70,136,112),(40,111,137,69),(41,50,90,99),(42,98,91,49),(43,48,92,97),(44,96,93,47),(45,46,94,95),(51,60,100,89),(52,88,81,59),(53,58,82,87),(54,86,83,57),(55,56,84,85)])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 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 | 2 | 2 | 2 | 2 | 20 | 20 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | D4×D5 | D4⋊2D5 |
| kernel | C23.45D20 | C10.10C42 | C2×D10⋊C4 | C10×C22⋊C4 | C23×Dic5 | C22×C5⋊D4 | C2×C5⋊D4 | C2×Dic5 | C22×C10 | C2×C22⋊C4 | C2×C10 | C22×C4 | C24 | C23 | C23 | C23 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 4 | 2 | 4 | 4 | 2 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of C23.45D20 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 14 | 27 | 0 | 0 |
| 0 | 14 | 30 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 2 | 9 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 27 | 14 | 0 | 0 |
| 0 | 30 | 14 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,9,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[32,0,0,0,0,0,14,14,0,0,0,27,30,0,0,0,0,0,32,2,0,0,0,0,9],[9,0,0,0,0,0,27,30,0,0,0,14,14,0,0,0,0,0,32,0,0,0,0,0,32] >;
C23.45D20 in GAP, Magma, Sage, TeX
C_2^3._{45}D_{20} % in TeX
G:=Group("C2^3.45D20"); // GroupNames label
G:=SmallGroup(320,585);
// by ID
G=gap.SmallGroup(320,585);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,422,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=b,f^2=c*b=b*c,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*a,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^9>;
// generators/relations