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G = D5×C2×C42order 320 = 26·5

Direct product of C2×C42 and D5

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

Aliases: D5×C2×C42, C209(C22×C4), C102(C2×C42), C52(C22×C42), (C4×C20)⋊54C22, (C2×C10).15C24, C10.23(C23×C4), Dic59(C22×C4), (C2×C20).874C23, (C4×Dic5)⋊83C22, D10.50(C22×C4), (C22×C4).466D10, C22.12(C23×D5), C23.312(C22×D5), (C22×C20).563C22, (C22×C10).377C23, (C2×Dic5).368C23, (C23×D5).143C22, (C22×D5).289C23, (C22×Dic5).287C22, (C2×C4×C20)⋊16C2, (C2×C20)⋊44(C2×C4), C2.1(D5×C22×C4), (C2×C4×Dic5)⋊39C2, C22.67(C2×C4×D5), (D5×C22×C4).38C2, (C2×Dic5)⋊38(C2×C4), (C2×C4×D5).421C22, (C2×C4).816(C22×D5), (C2×C10).246(C22×C4), (C22×D5).140(C2×C4), SmallGroup(320,1143)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C2×C42
C1C5C10C2×C10C22×D5C23×D5D5×C22×C4 — D5×C2×C42
C5 — D5×C2×C42
C1C2×C42

Generators and relations for D5×C2×C42
 G = < a,b,c,d,e | a2=b4=c4=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1278 in 498 conjugacy classes, 303 normal (10 characteristic)
C1, C2 [×7], C2 [×8], C4 [×12], C4 [×12], C22, C22 [×6], C22 [×28], C5, C2×C4 [×18], C2×C4 [×66], C23, C23 [×14], D5 [×8], C10 [×7], C42 [×4], C42 [×12], C22×C4 [×3], C22×C4 [×39], C24, Dic5 [×12], C20 [×12], D10 [×28], C2×C10, C2×C10 [×6], C2×C42, C2×C42 [×11], C23×C4 [×3], C4×D5 [×48], C2×Dic5 [×18], C2×C20 [×18], C22×D5 [×14], C22×C10, C22×C42, C4×Dic5 [×12], C4×C20 [×4], C2×C4×D5 [×36], C22×Dic5 [×3], C22×C20 [×3], C23×D5, D5×C42 [×8], C2×C4×Dic5 [×3], C2×C4×C20, D5×C22×C4 [×3], D5×C2×C42
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], D5, C42 [×16], C22×C4 [×42], C24, D10 [×7], C2×C42 [×12], C23×C4 [×3], C4×D5 [×12], C22×D5 [×7], C22×C42, C2×C4×D5 [×18], C23×D5, D5×C42 [×4], D5×C22×C4 [×3], D5×C2×C42

Smallest permutation representation of D5×C2×C42
On 160 points
Generators in S160
(1 89)(2 90)(3 86)(4 87)(5 88)(6 81)(7 82)(8 83)(9 84)(10 85)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)
(1 74 19 69)(2 75 20 70)(3 71 16 66)(4 72 17 67)(5 73 18 68)(6 76 11 61)(7 77 12 62)(8 78 13 63)(9 79 14 64)(10 80 15 65)(21 41 36 56)(22 42 37 57)(23 43 38 58)(24 44 39 59)(25 45 40 60)(26 46 31 51)(27 47 32 52)(28 48 33 53)(29 49 34 54)(30 50 35 55)(81 151 96 146)(82 152 97 147)(83 153 98 148)(84 154 99 149)(85 155 100 150)(86 156 91 141)(87 157 92 142)(88 158 93 143)(89 159 94 144)(90 160 95 145)(101 121 116 136)(102 122 117 137)(103 123 118 138)(104 124 119 139)(105 125 120 140)(106 126 111 131)(107 127 112 132)(108 128 113 133)(109 129 114 134)(110 130 115 135)
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)(81 111 91 101)(82 112 92 102)(83 113 93 103)(84 114 94 104)(85 115 95 105)(86 116 96 106)(87 117 97 107)(88 118 98 108)(89 119 99 109)(90 120 100 110)(121 151 131 141)(122 152 132 142)(123 153 133 143)(124 154 134 144)(125 155 135 145)(126 156 136 146)(127 157 137 147)(128 158 138 148)(129 159 139 149)(130 160 140 150)
(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 13)(2 12)(3 11)(4 15)(5 14)(6 16)(7 20)(8 19)(9 18)(10 17)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)(41 51)(42 55)(43 54)(44 53)(45 52)(46 56)(47 60)(48 59)(49 58)(50 57)(61 71)(62 75)(63 74)(64 73)(65 72)(66 76)(67 80)(68 79)(69 78)(70 77)(81 91)(82 95)(83 94)(84 93)(85 92)(86 96)(87 100)(88 99)(89 98)(90 97)(101 111)(102 115)(103 114)(104 113)(105 112)(106 116)(107 120)(108 119)(109 118)(110 117)(121 131)(122 135)(123 134)(124 133)(125 132)(126 136)(127 140)(128 139)(129 138)(130 137)(141 151)(142 155)(143 154)(144 153)(145 152)(146 156)(147 160)(148 159)(149 158)(150 157)

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

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

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

128 conjugacy classes

class 1 2A···2G2H···2O4A···4X4Y···4AV5A5B10A···10N20A···20AV
order12···22···24···44···45510···1020···20
size11···15···51···15···5222···22···2

128 irreducible representations

dim1111112222
type++++++++
imageC1C2C2C2C2C4D5D10D10C4×D5
kernelD5×C2×C42D5×C42C2×C4×Dic5C2×C4×C20D5×C22×C4C2×C4×D5C2×C42C42C22×C4C2×C4
# reps183134828648

Matrix representation of D5×C2×C42 in GL4(𝔽41) generated by

40000
04000
0010
0001
,
1000
03200
00320
00032
,
40000
03200
0010
0001
,
1000
0100
004036
004035
,
40000
0100
00400
00401
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[40,0,0,0,0,32,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,40,0,0,36,35],[40,0,0,0,0,1,0,0,0,0,40,40,0,0,0,1] >;

D5×C2×C42 in GAP, Magma, Sage, TeX

D_5\times C_2\times C_4^2
% in TeX

G:=Group("D5xC2xC4^2");
// GroupNames label

G:=SmallGroup(320,1143);
// by ID

G=gap.SmallGroup(320,1143);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,80,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^5=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

׿
×
𝔽