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G = C22×Dic10order 160 = 25·5

Direct product of C22 and Dic10

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22×Dic10, C10.1C24, C20.34C23, C23.33D10, Dic5.1C23, (C2×C10)⋊4Q8, C101(C2×Q8), C51(C22×Q8), (C2×C4).86D10, (C22×C4).8D5, C2.3(C23×D5), (C22×C20).8C2, C4.32(C22×D5), (C2×C20).95C22, (C2×C10).62C23, (C22×Dic5).6C2, C22.28(C22×D5), (C22×C10).43C22, (C2×Dic5).45C22, SmallGroup(160,213)

Series: Derived Chief Lower central Upper central

C1C10 — C22×Dic10
C1C5C10Dic5C2×Dic5C22×Dic5 — C22×Dic10
C5C10 — C22×Dic10
C1C23C22×C4

Generators and relations for C22×Dic10
 G = < a,b,c,d | a2=b2=c20=1, d2=c10, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 360 in 156 conjugacy classes, 105 normal (9 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×8], C22 [×7], C5, C2×C4 [×6], C2×C4 [×12], Q8 [×16], C23, C10, C10 [×6], C22×C4, C22×C4 [×2], C2×Q8 [×12], Dic5 [×8], C20 [×4], C2×C10 [×7], C22×Q8, Dic10 [×16], C2×Dic5 [×12], C2×C20 [×6], C22×C10, C2×Dic10 [×12], C22×Dic5 [×2], C22×C20, C22×Dic10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, Dic10 [×4], C22×D5 [×7], C2×Dic10 [×6], C23×D5, C22×Dic10

Smallest permutation representation of C22×Dic10
Regular action on 160 points
Generators in S160
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 154)(22 155)(23 156)(24 157)(25 158)(26 159)(27 160)(28 141)(29 142)(30 143)(31 144)(32 145)(33 146)(34 147)(35 148)(36 149)(37 150)(38 151)(39 152)(40 153)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 101)(77 102)(78 103)(79 104)(80 105)(81 137)(82 138)(83 139)(84 140)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 21)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(41 154)(42 155)(43 156)(44 157)(45 158)(46 159)(47 160)(48 141)(49 142)(50 143)(51 144)(52 145)(53 146)(54 147)(55 148)(56 149)(57 150)(58 151)(59 152)(60 153)(61 131)(62 132)(63 133)(64 134)(65 135)(66 136)(67 137)(68 138)(69 139)(70 140)(71 121)(72 122)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 129)(80 130)(81 112)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)(89 120)(90 101)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(97 108)(98 109)(99 110)(100 111)
(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)
(1 106 11 116)(2 105 12 115)(3 104 13 114)(4 103 14 113)(5 102 15 112)(6 101 16 111)(7 120 17 110)(8 119 18 109)(9 118 19 108)(10 117 20 107)(21 82 31 92)(22 81 32 91)(23 100 33 90)(24 99 34 89)(25 98 35 88)(26 97 36 87)(27 96 37 86)(28 95 38 85)(29 94 39 84)(30 93 40 83)(41 68 51 78)(42 67 52 77)(43 66 53 76)(44 65 54 75)(45 64 55 74)(46 63 56 73)(47 62 57 72)(48 61 58 71)(49 80 59 70)(50 79 60 69)(121 141 131 151)(122 160 132 150)(123 159 133 149)(124 158 134 148)(125 157 135 147)(126 156 136 146)(127 155 137 145)(128 154 138 144)(129 153 139 143)(130 152 140 142)

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

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

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

C22×Dic10 is a maximal subgroup of
(C2×C20)⋊Q8  (C2×Dic5)⋊Q8  (C2×C4).20D20  Dic1014D4  C22⋊Dic20  (C2×C20)⋊10Q8  C23⋊Dic10  (C2×Dic5)⋊6Q8  (C2×C4)⋊Dic10  C4.(C2×D20)  Dic1017D4  Dic10.37D4  C23.46D20  C42.87D10  C42.92D10  Dic1023D4  Dic1019D4  Dic1021D4  C10.792- 1+4  C10.1052- 1+4  C22×Q8×D5
C22×Dic10 is a maximal quotient of
C42.274D10  C232Dic10  C10.12- 1+4  C42.88D10  C42.90D10  D45Dic10  D46Dic10  Q85Dic10  Q86Dic10

52 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4L5A5B10A···10N20A···20P
order12···244444···45510···1020···20
size11···1222210···10222···22···2

52 irreducible representations

dim111122222
type++++-+++-
imageC1C2C2C2Q8D5D10D10Dic10
kernelC22×Dic10C2×Dic10C22×Dic5C22×C20C2×C10C22×C4C2×C4C23C22
# reps112214212216

Matrix representation of C22×Dic10 in GL4(𝔽41) generated by

40000
04000
0010
0001
,
40000
0100
00400
00040
,
40000
0100
002839
00216
,
40000
04000
00289
001313
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,28,2,0,0,39,16],[40,0,0,0,0,40,0,0,0,0,28,13,0,0,9,13] >;

C22×Dic10 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{10}
% in TeX

G:=Group("C2^2xDic10");
// GroupNames label

G:=SmallGroup(160,213);
// by ID

G=gap.SmallGroup(160,213);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,579,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^20=1,d^2=c^10,a*b=b*a,a*c=c*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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