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G = C23.23D20order 320 = 26·5

2nd non-split extension by C23 of D20 acting via D20/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.23D20, C4○D208C4, (C22×C8)⋊5D5, (C22×C40)⋊4C2, D20.37(C2×C4), C20.411(C2×D4), (C2×C20).403D4, (C2×C4).172D20, (C2×C8).295D10, D205C444C2, C10.16(C4○D8), C22.54(C2×D20), C2.5(D407C2), C20.44D444C2, (C2×C40).356C22, (C2×C20).767C23, C20.172(C22×C4), Dic10.39(C2×C4), (C22×C4).429D10, (C22×C10).139D4, C55(C23.24D4), C4.55(D10⋊C4), C20.113(C22⋊C4), (C2×D20).205C22, C23.21D102C2, C4⋊Dic5.282C22, C22.6(D10⋊C4), (C22×C20).542C22, (C2×Dic10).225C22, C4.71(C2×C4×D5), (C2×C4○D20).5C2, (C2×C4).117(C4×D5), C4.104(C2×C5⋊D4), (C2×C20).408(C2×C4), (C2×C10).157(C2×D4), C10.94(C2×C22⋊C4), C2.25(C2×D10⋊C4), (C2×C4).255(C5⋊D4), (C2×C4).715(C22×D5), (C2×C10).128(C22⋊C4), SmallGroup(320,740)

Series: Derived Chief Lower central Upper central

C1C20 — C23.23D20
C1C5C10C20C2×C20C2×D20C2×C4○D20 — C23.23D20
C5C10C20 — C23.23D20
C1C2×C4C22×C4C22×C8

Generators and relations for C23.23D20
 G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd19 >

Subgroups: 622 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×7], D4 [×7], Q8 [×3], C23, C23, D5 [×2], C10, C10 [×2], C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×2], C2×Q8, C4○D4 [×6], Dic5 [×4], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C42⋊C2, C22×C8, C2×C4○D4, C40 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5, C22×C10, C23.24D4, C4×Dic5, C4⋊Dic5 [×2], C23.D5, C2×C40 [×2], C2×C40 [×2], C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, C20.44D4 [×2], D205C4 [×2], C23.21D10, C22×C40, C2×C4○D20, C23.23D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4○D8 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.24D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, D407C2 [×2], C2×D10⋊C4, C23.23D20

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

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

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

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

92 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4L5A5B8A···8H10A···10N20A···20P40A···40AF
order122222224444444···4558···810···1020···2040···40
size111122202011112220···20222···22···22···22···2

92 irreducible representations

dim111111122222222222
type+++++++++++++
imageC1C2C2C2C2C2C4D4D4D5D10D10C4○D8C4×D5D20C5⋊D4D20D407C2
kernelC23.23D20C20.44D4D205C4C23.21D10C22×C40C2×C4○D20C4○D20C2×C20C22×C10C22×C8C2×C8C22×C4C10C2×C4C2×C4C2×C4C23C2
# reps1221118312428848432

Matrix representation of C23.23D20 in GL4(𝔽41) generated by

1000
0100
00236
003518
,
40000
04000
00400
00040
,
1000
0100
00400
00040
,
182400
382100
00829
001218
,
184000
382300
00298
001812
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,23,35,0,0,6,18],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[18,38,0,0,24,21,0,0,0,0,8,12,0,0,29,18],[18,38,0,0,40,23,0,0,0,0,29,18,0,0,8,12] >;

C23.23D20 in GAP, Magma, Sage, TeX

C_2^3._{23}D_{20}
% in TeX

G:=Group("C2^3.23D20");
// GroupNames label

G:=SmallGroup(320,740);
// by ID

G=gap.SmallGroup(320,740);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,142,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^19>;
// generators/relations

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