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G = C23.42D20order 320 = 26·5

13rd non-split extension by C23 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.42D20, C24.45D10, C23.50(C4×D5), (C22×C10).61D4, (C22×C4).23D10, C22.41(C2×D20), (C22×Dic5)⋊12C4, C23.50(C5⋊D4), C53(C23.34D4), (C23×Dic5).3C2, (C22×C20).22C22, (C23×C10).26C22, C23.276(C22×D5), C10.10C4210C2, C10.44(C42⋊C2), C22.42(D42D5), (C22×C10).318C23, C2.3(C22.D20), C22.22(D10⋊C4), C10.70(C22.D4), C2.1(C23.18D10), C2.12(C23.11D10), (C22×Dic5).206C22, (C2×C22⋊C4).5D5, C22.122(C2×C4×D5), C2.7(C2×D10⋊C4), (C2×C10).149(C2×D4), (C10×C22⋊C4).6C2, C10.75(C2×C22⋊C4), C22.46(C2×C5⋊D4), (C2×C23.D5).5C2, (C2×C10).139(C4○D4), (C2×C10).77(C22⋊C4), (C22×C10).114(C2×C4), (C2×C10).205(C22×C4), (C2×Dic5).148(C2×C4), SmallGroup(320,570)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C23.42D20
C1C5C10C2×C10C22×C10C22×Dic5C23×Dic5 — C23.42D20
C5C2×C10 — C23.42D20
C1C23C2×C22⋊C4

Generators and relations for C23.42D20
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 686 in 218 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C22 [×3], C22 [×8], C22 [×12], C5, C2×C4 [×28], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C22⋊C4 [×4], C22×C4 [×2], C22×C4 [×12], C24, Dic5 [×6], C20 [×2], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2.C42 [×4], C2×C22⋊C4, C2×C22⋊C4, C23×C4, C2×Dic5 [×4], C2×Dic5 [×18], C2×C20 [×6], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.34D4, C23.D5 [×2], C5×C22⋊C4 [×2], C22×Dic5 [×8], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C10.10C42 [×4], C2×C23.D5, C10×C22⋊C4, C23×Dic5, C23.42D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], D10 [×3], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.34D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, D42D5 [×4], C2×C5⋊D4, C23.11D10 [×2], C22.D20 [×2], C2×D10⋊C4, C23.18D10 [×2], C23.42D20

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

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4L4M4N4O4P5A5B10A···10N10O···10V20A···20P
order12···2222244444···444445510···1010···1020···20
size11···12222444410···1020202020222···24···44···4

68 irreducible representations

dim111111222222224
type++++++++++-
imageC1C2C2C2C2C4D4D5C4○D4D10D10C4×D5D20C5⋊D4D42D5
kernelC23.42D20C10.10C42C2×C23.D5C10×C22⋊C4C23×Dic5C22×Dic5C22×C10C2×C22⋊C4C2×C10C22×C4C24C23C23C23C22
# reps141118428428888

Matrix representation of C23.42D20 in GL5(𝔽41)

10000
01000
00100
00010
000040
,
400000
01000
00100
00010
00001
,
10000
01000
00100
000400
000040
,
90000
013200
0392500
00001
00010
,
320000
0121800
082900
000032
000320

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[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],[9,0,0,0,0,0,13,39,0,0,0,2,25,0,0,0,0,0,0,1,0,0,0,1,0],[32,0,0,0,0,0,12,8,0,0,0,18,29,0,0,0,0,0,0,32,0,0,0,32,0] >;

C23.42D20 in GAP, Magma, Sage, TeX

C_2^3._{42}D_{20}
% in TeX

G:=Group("C2^3.42D20");
// GroupNames label

G:=SmallGroup(320,570);
// by ID

G=gap.SmallGroup(320,570);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,422,387,58,12550]);
// Polycyclic

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

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