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G = C23⋊Dic10order 320 = 26·5

1st semidirect product of C23 and Dic10 acting via Dic10/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.5D10, C231Dic10, (C2×C20).48D4, (C22×C10)⋊3Q8, C52(C23⋊Q8), C10.36C22≀C2, (C2×Dic5).61D4, (C22×C4).27D10, C22.237(D4×D5), C2.6(C23⋊D10), (C22×Dic10)⋊3C2, C10.55(C22⋊Q8), C2.6(C20.48D4), C10.31(C4.4D4), C2.5(C20.17D4), C22.94(C4○D20), (C23×C10).30C22, (C22×C20).56C22, C22.44(C2×Dic10), C23.366(C22×D5), C10.10C4228C2, C22.92(D42D5), (C22×C10).322C23, C2.20(Dic5.5D4), (C22×Dic5).38C22, C2.20(Dic5.14D4), (C2×C10).32(C2×Q8), (C2×C10).316(C2×D4), (C2×C4).27(C5⋊D4), (C2×C22⋊C4).10D5, (C2×C23.D5).9C2, (C2×C10).76(C4○D4), (C10×C22⋊C4).12C2, C22.122(C2×C5⋊D4), SmallGroup(320,574)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C23⋊Dic10
C1C5C10C2×C10C22×C10C22×Dic5C22×Dic10 — C23⋊Dic10
C5C22×C10 — C23⋊Dic10
C1C23C2×C22⋊C4

Generators and relations for C23⋊Dic10
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=d10, eae-1=ab=ba, dad-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 710 in 202 conjugacy classes, 63 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×9], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×2], C2×C4 [×19], Q8 [×8], C23, C23 [×2], C23 [×6], C10 [×3], C10 [×4], C10 [×2], C22⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C2×Q8 [×6], C24, Dic5 [×6], C20 [×3], C2×C10 [×3], C2×C10 [×4], C2×C10 [×10], C2.C42 [×3], C2×C22⋊C4, C2×C22⋊C4 [×2], C22×Q8, Dic10 [×8], C2×Dic5 [×4], C2×Dic5 [×10], C2×C20 [×2], C2×C20 [×5], C22×C10, C22×C10 [×2], C22×C10 [×6], C23⋊Q8, C23.D5 [×4], C5×C22⋊C4 [×2], C2×Dic10 [×6], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C10.10C42, C10.10C42 [×2], C2×C23.D5 [×2], C10×C22⋊C4, C22×Dic10, C23⋊Dic10
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D5, C2×D4 [×3], C2×Q8, C4○D4 [×3], D10 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], Dic10 [×2], C5⋊D4 [×2], C22×D5, C23⋊Q8, C2×Dic10, C4○D20, D4×D5 [×2], D42D5 [×2], C2×C5⋊D4, Dic5.14D4 [×2], Dic5.5D4 [×2], C20.48D4, C20.17D4, C23⋊D10, C23⋊Dic10

Smallest permutation representation of C23⋊Dic10
On 160 points
Generators in S160
(2 23)(4 25)(6 27)(8 29)(10 31)(12 33)(14 35)(16 37)(18 39)(20 21)(41 117)(42 155)(43 119)(44 157)(45 101)(46 159)(47 103)(48 141)(49 105)(50 143)(51 107)(52 145)(53 109)(54 147)(55 111)(56 149)(57 113)(58 151)(59 115)(60 153)(61 138)(63 140)(65 122)(67 124)(69 126)(71 128)(73 130)(75 132)(77 134)(79 136)(81 154)(82 118)(83 156)(84 120)(85 158)(86 102)(87 160)(88 104)(89 142)(90 106)(91 144)(92 108)(93 146)(94 110)(95 148)(96 112)(97 150)(98 114)(99 152)(100 116)
(1 131)(2 132)(3 133)(4 134)(5 135)(6 136)(7 137)(8 138)(9 139)(10 140)(11 121)(12 122)(13 123)(14 124)(15 125)(16 126)(17 127)(18 128)(19 129)(20 130)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 117)(42 118)(43 119)(44 120)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(81 154)(82 155)(83 156)(84 157)(85 158)(86 159)(87 160)(88 141)(89 142)(90 143)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 151)(99 152)(100 153)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 21)(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)(61 138)(62 139)(63 140)(64 121)(65 122)(66 123)(67 124)(68 125)(69 126)(70 127)(71 128)(72 129)(73 130)(74 131)(75 132)(76 133)(77 134)(78 135)(79 136)(80 137)(101 158)(102 159)(103 160)(104 141)(105 142)(106 143)(107 144)(108 145)(109 146)(110 147)(111 148)(112 149)(113 150)(114 151)(115 152)(116 153)(117 154)(118 155)(119 156)(120 157)
(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 93 11 83)(2 92 12 82)(3 91 13 81)(4 90 14 100)(5 89 15 99)(6 88 16 98)(7 87 17 97)(8 86 18 96)(9 85 19 95)(10 84 20 94)(21 54 31 44)(22 53 32 43)(23 52 33 42)(24 51 34 41)(25 50 35 60)(26 49 36 59)(27 48 37 58)(28 47 38 57)(29 46 39 56)(30 45 40 55)(61 102 71 112)(62 101 72 111)(63 120 73 110)(64 119 74 109)(65 118 75 108)(66 117 76 107)(67 116 77 106)(68 115 78 105)(69 114 79 104)(70 113 80 103)(121 156 131 146)(122 155 132 145)(123 154 133 144)(124 153 134 143)(125 152 135 142)(126 151 136 141)(127 150 137 160)(128 149 138 159)(129 148 139 158)(130 147 140 157)

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

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

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

62 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E···4L5A5B10A···10N10O···10V20A···20P
order12···22244444···45510···1010···1020···20
size11···144444420···20222···24···44···4

62 irreducible representations

dim11111222222222244
type+++++++-+++-+-
imageC1C2C2C2C2D4D4Q8D5C4○D4D10D10C5⋊D4Dic10C4○D20D4×D5D42D5
kernelC23⋊Dic10C10.10C42C2×C23.D5C10×C22⋊C4C22×Dic10C2×Dic5C2×C20C22×C10C2×C22⋊C4C2×C10C22×C4C24C2×C4C23C22C22C22
# reps13211422264288844

Matrix representation of C23⋊Dic10 in GL6(𝔽41)

4000000
4010000
001000
0014000
000010
00002640
,
4000000
0400000
001000
000100
0000400
0000040
,
4000000
0400000
0040000
0004000
000010
000001
,
1390000
0400000
0013900
0004000
0000210
000012
,
4020000
010000
001000
000100
00002639
00003115

G:=sub<GL(6,GF(41))| [40,40,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,26,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,1,0,0,0,0,0,39,40,0,0,0,0,0,0,21,1,0,0,0,0,0,2],[40,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,31,0,0,0,0,39,15] >;

C23⋊Dic10 in GAP, Magma, Sage, TeX

C_2^3\rtimes {\rm Dic}_{10}
% in TeX

G:=Group("C2^3:Dic10");
// GroupNames label

G:=SmallGroup(320,574);
// by ID

G=gap.SmallGroup(320,574);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,254,387,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=d^10,e*a*e^-1=a*b=b*a,d*a*d^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽