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G = C22⋊C4×C2×C10  order 320 = 26·5

Direct product of C2×C10 and C22⋊C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C22⋊C4×C2×C10, C24⋊5C20, C25.3C10, (C23×C4)⋊2C10, (C23×C20)⋊5C2, C23⋊5(C2×C20), (C2×C20)⋊13C23, (C23×C10)⋊11C4, (C24×C10).2C2, C2.1(C23×C20), C23.58(C5×D4), C24.28(C2×C10), C10.74(C23×C4), C22⋊2(C22×C20), C22.57(D4×C10), (C2×C10).332C24, (C22×C20)⋊57C22, C10.177(C22×D4), (C22×C10).219D4, C22.5(C23×C10), C23.65(C22×C10), (C23×C10).88C22, (C22×C10).251C23, C2.1(D4×C2×C10), (C2×C4)⋊3(C22×C10), (C2×C10)⋊11(C22×C4), (C22×C10)⋊23(C2×C4), (C22×C4)⋊15(C2×C10), (C2×C10).679(C2×D4), SmallGroup(320,1514)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C22⋊C4×C2×C10
Chief series
C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C22⋊C4 — C10×C22⋊C4 — C22⋊C4×C2×C10
Lower central
C1 — C2 — C22⋊C4×C2×C10
Upper central
C1 — C23×C10 — C22⋊C4×C2×C10

Generators and relations for C22⋊C4×C2×C10
 G = < a,b,c,d,e | a2=b10=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 1010 in 674 conjugacy classes, 338 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C10, C10, C10, C22⋊C4, C22×C4, C22×C4, C24, C24, C24, C20, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C23×C4, C25, C2×C20, C2×C20, C22×C10, C22×C10, C22×C22⋊C4, C5×C22⋊C4, C22×C20, C22×C20, C23×C10, C23×C10, C23×C10, C10×C22⋊C4, C23×C20, C24×C10, C22⋊C4×C2×C10
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C24, C20, C2×C10, C2×C22⋊C4, C23×C4, C22×D4, C2×C20, C5×D4, C22×C10, C22×C22⋊C4, C5×C22⋊C4, C22×C20, D4×C10, C23×C10, C10×C22⋊C4, C23×C20, D4×C2×C10, C22⋊C4×C2×C10

Smallest permutation representation of C22⋊C4×C2×C10
►On 160 points
Generators in S160
(1 86)(2 87)(3 88)(4 89)(5 90)(6 81)(7 82)(8 83)(9 84)(10 85)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 111)(19 112)(20 113)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 131)(28 132)(29 133)(30 134)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 121)(38 122)(39 123)(40 124)(41 96)(42 97)(43 98)(44 99)(45 100)(46 91)(47 92)(48 93)(49 94)(50 95)(51 108)(52 109)(53 110)(54 101)(55 102)(56 103)(57 104)(58 105)(59 106)(60 107)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 71)(70 72)(141 153)(142 154)(143 155)(144 156)(145 157)(146 158)(147 159)(148 160)(149 151)(150 152)

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

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

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

200 conjugacy classes

class 1 2A···2O2P···2W4A···4P5A5B5C5D10A···10BH10BI···10CN20A···20BL
order12···22···24···4555510···1010···1020···20
size11···12···22···211111···12···22···2

200 irreducible representations

dim111111111122
type+++++
imageC1C2C2C2C4C5C10C10C10C20D4C5×D4
kernelC22⋊C4×C2×C10C10×C22⋊C4C23×C20C24×C10C23×C10C22×C22⋊C4C2×C22⋊C4C23×C4C25C24C22×C10C23
# reps11221164488464832

Matrix representation of C22⋊C4×C2×C10 ►in GL5(𝔽41)

400000
01000
00100
000400
000040
,
10000
01000
004000
000370
000037
,
400000
01000
00100
00010
000040
,
10000
01000
00100
000400
000040
,
10000
032000
00100
00001
00010

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,37,0,0,0,0,0,37],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40],[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],[1,0,0,0,0,0,32,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
 

C22⋊C4×C2×C10 in GAP, Magma, Sage, TeX 

C_2^2\rtimes C_4\times C_2\times C_{10}
 
% in TeX
 
G:=Group("C2^2:C4xC2xC10");
 
// GroupNames label
 
G:=SmallGroup(320,1514);
 
// by ID
 
G=gap.SmallGroup(320,1514);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,d*e=e*d>;
 
// generators/relations
 

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