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G = C23.45D20  order 320 = 26·5

16th non-split extension by C23 of D20 acting via D20/D10=C2

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

Derived series
C1 — C2×C10 — C23.45D20
Chief series
C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C22×C5⋊D4 — C23.45D20
Lower central
C5 — C2×C10 — C23.45D20
Upper central
C1 — C23 — C2×C22⋊C4

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, C2, C2, C4, C22, C22, C22, C5, 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, C23×C4, C22×D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C23.23D4, D10⋊C4, C5×C22⋊C4, C22×Dic5, C22×Dic5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C23×D5, C23×C10, C10.10C42, C2×D10⋊C4, C10×C22⋊C4, C23×Dic5, C22×C5⋊D4, C23.45D20
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, C4○D4, D10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C4×D5, D20, C5⋊D4, C22×D5, C23.23D4, D10⋊C4, C2×C4×D5, C2×D20, D4×D5, D4⋊2D5, C2×C5⋊D4, Dic5⋊4D4, C22⋊D20, C22.D20, C2×D10⋊C4, Dic5⋊D4, C23.45D20

Smallest permutation representation of C23.45D20
►On 160 points
Generators in S160
(1 57)(2 78)(3 59)(4 80)(5 41)(6 62)(7 43)(8 64)(9 45)(10 66)(11 47)(12 68)(13 49)(14 70)(15 51)(16 72)(17 53)(18 74)(19 55)(20 76)(21 77)(22 58)(23 79)(24 60)(25 61)(26 42)(27 63)(28 44)(29 65)(30 46)(31 67)(32 48)(33 69)(34 50)(35 71)(36 52)(37 73)(38 54)(39 75)(40 56)(81 139)(82 154)(83 121)(84 156)(85 123)(86 158)(87 125)(88 160)(89 127)(90 142)(91 129)(92 144)(93 131)(94 146)(95 133)(96 148)(97 135)(98 150)(99 137)(100 152)(101 124)(102 159)(103 126)(104 141)(105 128)(106 143)(107 130)(108 145)(109 132)(110 147)(111 134)(112 149)(113 136)(114 151)(115 138)(116 153)(117 140)(118 155)(119 122)(120 157)

(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)

(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 96)(22 97)(23 98)(24 99)(25 100)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 138)(42 139)(43 140)(44 121)(45 122)(46 123)(47 124)(48 125)(49 126)(50 127)(51 128)(52 129)(53 130)(54 131)(55 132)(56 133)(57 134)(58 135)(59 136)(60 137)(61 152)(62 153)(63 154)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 141)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)

(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(81 116)(82 117)(83 118)(84 119)(85 120)(86 101)(87 102)(88 103)(89 104)(90 105)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(121 155)(122 156)(123 157)(124 158)(125 159)(126 160)(127 141)(128 142)(129 143)(130 144)(131 145)(132 146)(133 147)(134 148)(135 149)(136 150)(137 151)(138 152)(139 153)(140 154)

(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 10 101 110)(2 109 102 9)(3 8 103 108)(4 107 104 7)(5 6 105 106)(11 20 111 120)(12 119 112 19)(13 18 113 118)(14 117 114 17)(15 16 115 116)(21 30 86 95)(22 94 87 29)(23 28 88 93)(24 92 89 27)(25 26 90 91)(31 40 96 85)(32 84 97 39)(33 38 98 83)(34 82 99 37)(35 36 100 81)(41 62 128 143)(42 142 129 61)(43 80 130 141)(44 160 131 79)(45 78 132 159)(46 158 133 77)(47 76 134 157)(48 156 135 75)(49 74 136 155)(50 154 137 73)(51 72 138 153)(52 152 139 71)(53 70 140 151)(54 150 121 69)(55 68 122 149)(56 148 123 67)(57 66 124 147)(58 146 125 65)(59 64 126 145)(60 144 127 63)
 
G:=sub<Sym(160)| (1,57)(2,78)(3,59)(4,80)(5,41)(6,62)(7,43)(8,64)(9,45)(10,66)(11,47)(12,68)(13,49)(14,70)(15,51)(16,72)(17,53)(18,74)(19,55)(20,76)(21,77)(22,58)(23,79)(24,60)(25,61)(26,42)(27,63)(28,44)(29,65)(30,46)(31,67)(32,48)(33,69)(34,50)(35,71)(36,52)(37,73)(38,54)(39,75)(40,56)(81,139)(82,154)(83,121)(84,156)(85,123)(86,158)(87,125)(88,160)(89,127)(90,142)(91,129)(92,144)(93,131)(94,146)(95,133)(96,148)(97,135)(98,150)(99,137)(100,152)(101,124)(102,159)(103,126)(104,141)(105,128)(106,143)(107,130)(108,145)(109,132)(110,147)(111,134)(112,149)(113,136)(114,151)(115,138)(116,153)(117,140)(118,155)(119,122)(120,157), (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), (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,96)(22,97)(23,98)(24,99)(25,100)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,138)(42,139)(43,140)(44,121)(45,122)(46,123)(47,124)(48,125)(49,126)(50,127)(51,128)(52,129)(53,130)(54,131)(55,132)(56,133)(57,134)(58,135)(59,136)(60,137)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(81,116)(82,117)(83,118)(84,119)(85,120)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(121,155)(122,156)(123,157)(124,158)(125,159)(126,160)(127,141)(128,142)(129,143)(130,144)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,151)(138,152)(139,153)(140,154), (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), (1,10,101,110)(2,109,102,9)(3,8,103,108)(4,107,104,7)(5,6,105,106)(11,20,111,120)(12,119,112,19)(13,18,113,118)(14,117,114,17)(15,16,115,116)(21,30,86,95)(22,94,87,29)(23,28,88,93)(24,92,89,27)(25,26,90,91)(31,40,96,85)(32,84,97,39)(33,38,98,83)(34,82,99,37)(35,36,100,81)(41,62,128,143)(42,142,129,61)(43,80,130,141)(44,160,131,79)(45,78,132,159)(46,158,133,77)(47,76,134,157)(48,156,135,75)(49,74,136,155)(50,154,137,73)(51,72,138,153)(52,152,139,71)(53,70,140,151)(54,150,121,69)(55,68,122,149)(56,148,123,67)(57,66,124,147)(58,146,125,65)(59,64,126,145)(60,144,127,63)>;
 
G:=Group( (1,57)(2,78)(3,59)(4,80)(5,41)(6,62)(7,43)(8,64)(9,45)(10,66)(11,47)(12,68)(13,49)(14,70)(15,51)(16,72)(17,53)(18,74)(19,55)(20,76)(21,77)(22,58)(23,79)(24,60)(25,61)(26,42)(27,63)(28,44)(29,65)(30,46)(31,67)(32,48)(33,69)(34,50)(35,71)(36,52)(37,73)(38,54)(39,75)(40,56)(81,139)(82,154)(83,121)(84,156)(85,123)(86,158)(87,125)(88,160)(89,127)(90,142)(91,129)(92,144)(93,131)(94,146)(95,133)(96,148)(97,135)(98,150)(99,137)(100,152)(101,124)(102,159)(103,126)(104,141)(105,128)(106,143)(107,130)(108,145)(109,132)(110,147)(111,134)(112,149)(113,136)(114,151)(115,138)(116,153)(117,140)(118,155)(119,122)(120,157), (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), (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,96)(22,97)(23,98)(24,99)(25,100)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,138)(42,139)(43,140)(44,121)(45,122)(46,123)(47,124)(48,125)(49,126)(50,127)(51,128)(52,129)(53,130)(54,131)(55,132)(56,133)(57,134)(58,135)(59,136)(60,137)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(81,116)(82,117)(83,118)(84,119)(85,120)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(121,155)(122,156)(123,157)(124,158)(125,159)(126,160)(127,141)(128,142)(129,143)(130,144)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,151)(138,152)(139,153)(140,154), (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), (1,10,101,110)(2,109,102,9)(3,8,103,108)(4,107,104,7)(5,6,105,106)(11,20,111,120)(12,119,112,19)(13,18,113,118)(14,117,114,17)(15,16,115,116)(21,30,86,95)(22,94,87,29)(23,28,88,93)(24,92,89,27)(25,26,90,91)(31,40,96,85)(32,84,97,39)(33,38,98,83)(34,82,99,37)(35,36,100,81)(41,62,128,143)(42,142,129,61)(43,80,130,141)(44,160,131,79)(45,78,132,159)(46,158,133,77)(47,76,134,157)(48,156,135,75)(49,74,136,155)(50,154,137,73)(51,72,138,153)(52,152,139,71)(53,70,140,151)(54,150,121,69)(55,68,122,149)(56,148,123,67)(57,66,124,147)(58,146,125,65)(59,64,126,145)(60,144,127,63) );
 
G=PermutationGroup([[(1,57),(2,78),(3,59),(4,80),(5,41),(6,62),(7,43),(8,64),(9,45),(10,66),(11,47),(12,68),(13,49),(14,70),(15,51),(16,72),(17,53),(18,74),(19,55),(20,76),(21,77),(22,58),(23,79),(24,60),(25,61),(26,42),(27,63),(28,44),(29,65),(30,46),(31,67),(32,48),(33,69),(34,50),(35,71),(36,52),(37,73),(38,54),(39,75),(40,56),(81,139),(82,154),(83,121),(84,156),(85,123),(86,158),(87,125),(88,160),(89,127),(90,142),(91,129),(92,144),(93,131),(94,146),(95,133),(96,148),(97,135),(98,150),(99,137),(100,152),(101,124),(102,159),(103,126),(104,141),(105,128),(106,143),(107,130),(108,145),(109,132),(110,147),(111,134),(112,149),(113,136),(114,151),(115,138),(116,153),(117,140),(118,155),(119,122),(120,157)], [(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)], [(1,111),(2,112),(3,113),(4,114),(5,115),(6,116),(7,117),(8,118),(9,119),(10,120),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,107),(18,108),(19,109),(20,110),(21,96),(22,97),(23,98),(24,99),(25,100),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,138),(42,139),(43,140),(44,121),(45,122),(46,123),(47,124),(48,125),(49,126),(50,127),(51,128),(52,129),(53,130),(54,131),(55,132),(56,133),(57,134),(58,135),(59,136),(60,137),(61,152),(62,153),(63,154),(64,155),(65,156),(66,157),(67,158),(68,159),(69,160),(70,141),(71,142),(72,143),(73,144),(74,145),(75,146),(76,147),(77,148),(78,149),(79,150),(80,151)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(81,116),(82,117),(83,118),(84,119),(85,120),(86,101),(87,102),(88,103),(89,104),(90,105),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,115),(121,155),(122,156),(123,157),(124,158),(125,159),(126,160),(127,141),(128,142),(129,143),(130,144),(131,145),(132,146),(133,147),(134,148),(135,149),(136,150),(137,151),(138,152),(139,153),(140,154)], [(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)], [(1,10,101,110),(2,109,102,9),(3,8,103,108),(4,107,104,7),(5,6,105,106),(11,20,111,120),(12,119,112,19),(13,18,113,118),(14,117,114,17),(15,16,115,116),(21,30,86,95),(22,94,87,29),(23,28,88,93),(24,92,89,27),(25,26,90,91),(31,40,96,85),(32,84,97,39),(33,38,98,83),(34,82,99,37),(35,36,100,81),(41,62,128,143),(42,142,129,61),(43,80,130,141),(44,160,131,79),(45,78,132,159),(46,158,133,77),(47,76,134,157),(48,156,135,75),(49,74,136,155),(50,154,137,73),(51,72,138,153),(52,152,139,71),(53,70,140,151),(54,150,121,69),(55,68,122,149),(56,148,123,67),(57,66,124,147),(58,146,125,65),(59,64,126,145),(60,144,127,63)]])
 

68 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4L4M4N5A5B10A···10N10O···10V20A···20P
order12···222222244444···4445510···1010···1020···20
size11···122222020444410···102020222···24···44···4

68 irreducible representations

dim111111122222222244
type+++++++++++++-
imageC1C2C2C2C2C2C4D4D4D5C4○D4D10D10C4×D5D20C5⋊D4D4×D5D4⋊2D5
kernelC23.45D20C10.10C42C2×D10⋊C4C10×C22⋊C4C23×Dic5C22×C5⋊D4C2×C5⋊D4C2×Dic5C22×C10C2×C22⋊C4C2×C10C22×C4C24C23C23C23C22C22
# reps122111844244288844

Matrix representation of C23.45D20 ►in GL5(𝔽41)

10000
040000
004000
00019
000040
,
400000
040000
004000
000400
000040
,
10000
040000
004000
00010
00001
,
10000
01000
00100
000400
000040
,
320000
0142700
0143000
000320
00029
,
90000
0271400
0301400
000320
000032

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
 

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