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G = C23.13D20order 320 = 26·5

6th non-split extension by C23 of D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.13D20, C22⋊C87D5, C406C49C2, C405C45C2, (C2×C8).110D10, C207D4.8C2, (C2×C20).242D4, (C2×C4).120D20, D205C411C2, C10.10(C4○D8), (C22×C4).87D10, (C22×C10).57D4, C20.284(C4○D4), C2.15(C8⋊D10), C10.12(C8⋊C22), (C2×C40).121C22, (C2×C20).747C23, (C2×D20).14C22, C22.110(C2×D20), C51(C23.19D4), C4⋊Dic5.14C22, C4.108(D42D5), C2.12(D407C2), C23.21D101C2, (C22×C20).98C22, C10.19(C22.D4), C2.15(C22.D20), (C5×C22⋊C8)⋊9C2, (C2×C10).130(C2×D4), (C2×C4).692(C22×D5), SmallGroup(320,364)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C23.13D20
C1C5C10C20C2×C20C2×D20C207D4 — C23.13D20
C5C10C2×C20 — C23.13D20
C1C22C22×C4C22⋊C8

Generators and relations for C23.13D20
 G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, ab=ba, eae-1=ac=ca, dad-1=abc, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd19 >

Subgroups: 494 in 106 conjugacy classes, 39 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×4], C23, C23, D5, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C22×C4, C2×D4 [×2], Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C40 [×2], D20 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C23.19D4, C4×Dic5, C4⋊Dic5 [×3], D10⋊C4, C23.D5, C2×C40 [×2], C2×D20, C2×C5⋊D4, C22×C20, C406C4, C405C4, D205C4 [×2], C5×C22⋊C8, C23.21D10, C207D4, C23.13D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C4○D8, C8⋊C22, D20 [×2], C22×D5, C23.19D4, C2×D20, D42D5 [×2], C22.D20, D407C2, C8⋊D10, C23.13D20

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

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222244444444455888810···101010101020···202020202040···40
size1111440222220202020402244442···244442···244444···4

59 irreducible representations

dim11111112222222222444
type+++++++++++++++-+
imageC1C2C2C2C2C2C2D4D4D5C4○D4D10D10C4○D8D20D20D407C2C8⋊C22D42D5C8⋊D10
kernelC23.13D20C406C4C405C4D205C4C5×C22⋊C8C23.21D10C207D4C2×C20C22×C10C22⋊C8C20C2×C8C22×C4C10C2×C4C23C2C10C4C2
# reps111211111244244416144

Matrix representation of C23.13D20 in GL4(𝔽41) generated by

17700
352400
001631
00525
,
40000
04000
00400
00040
,
40000
04000
0010
0001
,
202700
12600
003233
0009
,
382300
5300
00320
00032
G:=sub<GL(4,GF(41))| [17,35,0,0,7,24,0,0,0,0,16,5,0,0,31,25],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[20,12,0,0,27,6,0,0,0,0,32,0,0,0,33,9],[38,5,0,0,23,3,0,0,0,0,32,0,0,0,0,32] >;

C23.13D20 in GAP, Magma, Sage, TeX

C_2^3._{13}D_{20}
% in TeX

G:=Group("C2^3.13D20");
// GroupNames label

G:=SmallGroup(320,364);
// by ID

G=gap.SmallGroup(320,364);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,555,142,1123,136,12550]);
// 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,d*a*d^-1=a*b*c,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

׿
×
𝔽