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G = C22⋊Dic20order 320 = 26·5

The semidirect product of C22 and Dic20 acting via Dic20/Dic10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C222Dic20, C23.41D20, Dic10.32D4, (C2×C10)⋊1Q16, (C2×C8).5D10, C4.124(D4×D5), (C2×C4).36D20, (C2×C20).47D4, C22⋊C8.4D5, C10.6(C2×Q16), (C2×Dic20)⋊3C2, C20.336(C2×D4), (C2×C40).5C22, C51(C22⋊Q16), C2.8(C2×Dic20), C10.12C22≀C2, C20.44D47C2, (C22×C4).89D10, (C22×C10).59D4, (C2×C20).749C23, C20.48D4.4C2, C22.112(C2×D20), C4⋊Dic5.16C22, C2.15(C22⋊D20), C2.15(C8.D10), C10.12(C8.C22), (C22×C20).55C22, (C22×Dic10).3C2, (C2×Dic10).15C22, (C5×C22⋊C8).6C2, (C2×C10).132(C2×D4), (C2×C4).694(C22×D5), SmallGroup(320,366)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C22⋊Dic20
C1C5C10C20C2×C20C2×Dic10C22×Dic10 — C22⋊Dic20
C5C10C2×C20 — C22⋊Dic20
C1C22C22×C4C22⋊C8

Generators and relations for C22⋊Dic20
 G = < a,b,c,d | a2=b2=c40=1, d2=c20, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 590 in 148 conjugacy classes, 47 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×10], Q8 [×12], C23, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], Q16 [×4], C22×C4, C22×C4, C2×Q8 [×8], Dic5 [×6], C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, Q8⋊C4 [×2], C22⋊Q8, C2×Q16 [×2], C22×Q8, C40 [×2], Dic10 [×4], Dic10 [×8], C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×2], C22×C10, C22⋊Q16, Dic20 [×4], C10.D4, C4⋊Dic5, C23.D5, C2×C40 [×2], C2×Dic10, C2×Dic10 [×2], C2×Dic10 [×5], C22×Dic5, C22×C20, C20.44D4 [×2], C5×C22⋊C8, C2×Dic20 [×2], C20.48D4, C22×Dic10, C22⋊Dic20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, Q16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×Q16, C8.C22, D20 [×2], C22×D5, C22⋊Q16, Dic20 [×2], C2×D20, D4×D5 [×2], C22⋊D20, C2×Dic20, C8.D10, C22⋊Dic20

Smallest permutation representation of C22⋊Dic20
On 160 points
Generators in S160
(1 21)(2 72)(3 23)(4 74)(5 25)(6 76)(7 27)(8 78)(9 29)(10 80)(11 31)(12 42)(13 33)(14 44)(15 35)(16 46)(17 37)(18 48)(19 39)(20 50)(22 52)(24 54)(26 56)(28 58)(30 60)(32 62)(34 64)(36 66)(38 68)(40 70)(41 61)(43 63)(45 65)(47 67)(49 69)(51 71)(53 73)(55 75)(57 77)(59 79)(81 101)(82 123)(83 103)(84 125)(85 105)(86 127)(87 107)(88 129)(89 109)(90 131)(91 111)(92 133)(93 113)(94 135)(95 115)(96 137)(97 117)(98 139)(99 119)(100 141)(102 143)(104 145)(106 147)(108 149)(110 151)(112 153)(114 155)(116 157)(118 159)(120 121)(122 142)(124 144)(126 146)(128 148)(130 150)(132 152)(134 154)(136 156)(138 158)(140 160)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(81 142)(82 143)(83 144)(84 145)(85 146)(86 147)(87 148)(88 149)(89 150)(90 151)(91 152)(92 153)(93 154)(94 155)(95 156)(96 157)(97 158)(98 159)(99 160)(100 121)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(113 134)(114 135)(115 136)(116 137)(117 138)(118 139)(119 140)(120 141)
(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)
(1 146 21 126)(2 145 22 125)(3 144 23 124)(4 143 24 123)(5 142 25 122)(6 141 26 121)(7 140 27 160)(8 139 28 159)(9 138 29 158)(10 137 30 157)(11 136 31 156)(12 135 32 155)(13 134 33 154)(14 133 34 153)(15 132 35 152)(16 131 36 151)(17 130 37 150)(18 129 38 149)(19 128 39 148)(20 127 40 147)(41 95 61 115)(42 94 62 114)(43 93 63 113)(44 92 64 112)(45 91 65 111)(46 90 66 110)(47 89 67 109)(48 88 68 108)(49 87 69 107)(50 86 70 106)(51 85 71 105)(52 84 72 104)(53 83 73 103)(54 82 74 102)(55 81 75 101)(56 120 76 100)(57 119 77 99)(58 118 78 98)(59 117 79 97)(60 116 80 96)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222244444444455888810···101010101020···202020202040···40
size1111222242020202040402244442···244442···244444···4

59 irreducible representations

dim1111112222222222444
type++++++++++-++++--+-
imageC1C2C2C2C2C2D4D4D4D5Q16D10D10D20D20Dic20C8.C22D4×D5C8.D10
kernelC22⋊Dic20C20.44D4C5×C22⋊C8C2×Dic20C20.48D4C22×Dic10Dic10C2×C20C22×C10C22⋊C8C2×C10C2×C8C22×C4C2×C4C23C22C10C4C2
# reps12121141124424416144

Matrix representation of C22⋊Dic20 in GL6(𝔽41)

100000
010000
001000
00394000
0000400
0000040
,
100000
010000
0040000
0004000
000010
000001
,
1400000
8340000
00404000
002100
0000024
00002924
,
350000
23380000
0040000
002100
00002435
00002117

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,39,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,8,0,0,0,0,40,34,0,0,0,0,0,0,40,2,0,0,0,0,40,1,0,0,0,0,0,0,0,29,0,0,0,0,24,24],[3,23,0,0,0,0,5,38,0,0,0,0,0,0,40,2,0,0,0,0,0,1,0,0,0,0,0,0,24,21,0,0,0,0,35,17] >;

C22⋊Dic20 in GAP, Magma, Sage, TeX

C_2^2\rtimes {\rm Dic}_{20}
% in TeX

G:=Group("C2^2:Dic20");
// GroupNames label

G:=SmallGroup(320,366);
// by ID

G=gap.SmallGroup(320,366);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,219,226,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^40=1,d^2=c^20,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
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