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G = D4×C22×C10  order 320 = 26·5

Direct product of C22×C10 and D4

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

Aliases: D4×C22×C10, C25⋊3C10, C20⋊4C24, C10.21C25, C4⋊(C23×C10), (C24×C10)⋊2C2, (C23×C4)⋊7C10, (C2×C10)⋊2C24, C24⋊8(C2×C10), C22⋊(C23×C10), (C2×C20)⋊17C23, (C23×C20)⋊16C2, C2.1(C24×C10), C23⋊3(C22×C10), (C22×C10)⋊8C23, (C23×C10)⋊17C22, (C22×C20)⋊66C22, (C2×C4)⋊4(C22×C10), (C22×C4)⋊19(C2×C10), SmallGroup(320,1629)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — D4×C22×C10
Chief series
C1 — C2 — C10 — C2×C10 — C5×D4 — D4×C10 — D4×C2×C10 — D4×C22×C10
Lower central
C1 — C2 — D4×C22×C10
Upper central
C1 — C23×C10 — D4×C22×C10

Subgroups: 1874 in 1362 conjugacy classes, 850 normal (10 characteristic)
C1, C2, C2 [×14], C2 [×16], C4 [×8], C22 [×51], C22 [×112], C5, C2×C4 [×28], D4 [×64], C23 [×71], C23 [×112], C10, C10 [×14], C10 [×16], C22×C4 [×14], C2×D4 [×112], C24, C24 [×28], C24 [×16], C20 [×8], C2×C10 [×51], C2×C10 [×112], C23×C4, C22×D4 [×28], C25 [×2], C2×C20 [×28], C5×D4 [×64], C22×C10 [×71], C22×C10 [×112], D4×C23, C22×C20 [×14], D4×C10 [×112], C23×C10, C23×C10 [×28], C23×C10 [×16], C23×C20, D4×C2×C10 [×28], C24×C10 [×2], D4×C22×C10

Quotients:
C1, C2 [×31], C22 [×155], C5, D4 [×8], C23 [×155], C10 [×31], C2×D4 [×28], C24 [×31], C2×C10 [×155], C22×D4 [×14], C25, C5×D4 [×8], C22×C10 [×155], D4×C23, D4×C10 [×28], C23×C10 [×31], D4×C2×C10 [×14], C24×C10, D4×C22×C10

Generators and relations
 G = < a,b,c,d,e | a2=b2=c10=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽41)

400000
01000
004000
000400
000040
,
400000
040000
004000
000400
000040
,
10000
01000
003100
000400
000040
,
400000
040000
004000
000402
000401
,
400000
040000
004000
000139
000040

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,40,0,0,0,2,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,39,40] >;
 

200 conjugacy classes

class 1 2A···2O2P···2AE4A···4H5A5B5C5D10A···10BH10BI···10DT20A···20AF
order12···22···24···4555510···1010···1020···20
size11···12···22···211111···12···22···2

200 irreducible representations

dim1111111122
type+++++
imageC1C2C2C2C5C10C10C10D4C5×D4
kernelD4×C22×C10C23×C20D4×C2×C10C24×C10D4×C23C23×C4C22×D4C25C22×C10C23
# reps11282441128832

In GAP, Magma, Sage, TeX 

D_4\times C_2^2\times C_{10}
 
% in TeX
 
G:=Group("D4xC2^2xC10");
 
// GroupNames label
 
G:=SmallGroup(320,1629);
 
// by ID
 
G=gap.SmallGroup(320,1629);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-5,-2,2269]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^10=d^4=e^2=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,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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×
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