Copied to
clipboard

?

G = D5×C22×C8  order 320 = 26·5

Direct product of C22×C8 and D5

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

Aliases: D5×C22×C8, C40⋊11C23, C20.65C24, C5⋊3(C23×C8), C10⋊3(C22×C8), (C2×C40)⋊47C22, (C22×C40)⋊17C2, C5⋊2C8⋊14C23, C4.64(C23×D5), C23.65(C4×D5), C10.49(C23×C4), (C4×D5).94C23, (C23×D5).20C4, (C2×C20).878C23, C20.203(C22×C4), D10.54(C22×C4), (C22×C4).469D10, (C22×Dic5).41C4, Dic5.56(C22×C4), (C22×C20).566C22, (C2×C4×D5).50C4, C4.119(C2×C4×D5), (C2×C10)⋊12(C2×C8), C2.2(D5×C22×C4), C22.74(C2×C4×D5), (C2×C4).186(C4×D5), (D5×C22×C4).39C2, (C2×C20).429(C2×C4), (C2×C5⋊2C8)⋊49C22, (C22×C5⋊2C8)⋊24C2, (C4×D5).101(C2×C4), (C2×C4×D5).422C22, (C2×C4).822(C22×D5), (C22×C10).169(C2×C4), (C2×C10).255(C22×C4), (C2×Dic5).208(C2×C4), (C22×D5).142(C2×C4), SmallGroup(320,1408)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — D5×C22×C8
Chief series
C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — D5×C22×C4 — D5×C22×C8
Lower central
C5 — D5×C22×C8

Subgroups: 862 in 338 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×8], C4, C4 [×3], C4 [×4], C22 [×7], C22 [×28], C5, C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×22], C23, C23 [×14], D5 [×8], C10, C10 [×6], C2×C8 [×6], C2×C8 [×22], C22×C4, C22×C4 [×13], C24, Dic5 [×4], C20, C20 [×3], D10 [×28], C2×C10 [×7], C22×C8, C22×C8 [×13], C23×C4, C5⋊2C8 [×4], C40 [×4], C4×D5 [×16], C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×14], C22×C10, C23×C8, C8×D5 [×16], C2×C5⋊2C8 [×6], C2×C40 [×6], C2×C4×D5 [×12], C22×Dic5, C22×C20, C23×D5, D5×C2×C8 [×12], C22×C5⋊2C8, C22×C40, D5×C22×C4, D5×C22×C8

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], D5, C2×C8 [×28], C22×C4 [×14], C24, D10 [×7], C22×C8 [×14], C23×C4, C4×D5 [×4], C22×D5 [×7], C23×C8, C8×D5 [×4], C2×C4×D5 [×6], C23×D5, D5×C2×C8 [×6], D5×C22×C4, D5×C22×C8

Generators and relations
 G = < a,b,c,d,e | a2=b2=c8=d5=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
(1 127)(2 128)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 105)(16 106)(17 85)(18 86)(19 87)(20 88)(21 81)(22 82)(23 83)(24 84)(25 93)(26 94)(27 95)(28 96)(29 89)(30 90)(31 91)(32 92)(33 101)(34 102)(35 103)(36 104)(37 97)(38 98)(39 99)(40 100)(41 117)(42 118)(43 119)(44 120)(45 113)(46 114)(47 115)(48 116)(49 133)(50 134)(51 135)(52 136)(53 129)(54 130)(55 131)(56 132)(57 141)(58 142)(59 143)(60 144)(61 137)(62 138)(63 139)(64 140)(65 149)(66 150)(67 151)(68 152)(69 145)(70 146)(71 147)(72 148)(73 157)(74 158)(75 159)(76 160)(77 153)(78 154)(79 155)(80 156)

(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 81)(8 82)(9 151)(10 152)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 121)(18 122)(19 123)(20 124)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 141)(38 142)(39 143)(40 144)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 89)(50 90)(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)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)

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

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

Matrix representation ►G ⊆ GL4(𝔽41) generated by

1000
04000
00400
00040
,
40000
04000
0010
0001
,
40000
04000
00380
00038
,
1000
0100
0001
004034
,
1000
0100
0010
003440
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,38,0,0,0,0,38],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,34],[1,0,0,0,0,1,0,0,0,0,1,34,0,0,0,40] >;
 

128 conjugacy classes

class 1 2A···2G2H···2O4A···4H4I···4P5A5B8A···8P8Q···8AF10A···10N20A···20P40A···40AF
order12···22···24···44···4558···88···810···1020···2040···40
size11···15···51···15···5221···15···52···22···22···2

128 irreducible representations

dim111111111222222
type++++++++
imageC1C2C2C2C2C4C4C4C8D5D10D10C4×D5C4×D5C8×D5
kernelD5×C22×C8D5×C2×C8C22×C5⋊2C8C22×C40D5×C22×C4C2×C4×D5C22×Dic5C23×D5C22×D5C22×C8C2×C8C22×C4C2×C4C23C22
# reps112111122232212212432

In GAP, Magma, Sage, TeX 

D_5\times C_2^2\times C_8
 
% in TeX
 
G:=Group("D5xC2^2xC8");
 
// GroupNames label
 
G:=SmallGroup(320,1408);
 
// by ID
 
G=gap.SmallGroup(320,1408);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,80,102,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^8=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
 

׿
×
⋊
ℤ
𝔽
○
≀