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## G = D5×C22×C8order 320 = 26·5

### Direct product of C22×C8 and D5

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
 Upper central C1 — C22×C8

Generators and relations for D5×C22×C8
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 >

Subgroups: 862 in 338 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, C23, D5, C10, C10, C2×C8, C2×C8, C22×C4, C22×C4, C24, Dic5, C20, C20, D10, C2×C10, C22×C8, C22×C8, C23×C4, C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C10, C23×C8, C8×D5, C2×C52C8, C2×C40, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, D5×C2×C8, C22×C52C8, C22×C40, D5×C22×C4, D5×C22×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C2×C8, C22×C4, C24, D10, C22×C8, C23×C4, C4×D5, C22×D5, C23×C8, C8×D5, C2×C4×D5, C23×D5, D5×C2×C8, D5×C22×C4, D5×C22×C8

Smallest permutation representation of D5×C22×C8
On 160 points
Generators in S160
(1 142)(2 143)(3 144)(4 137)(5 138)(6 139)(7 140)(8 141)(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)(41 125)(42 126)(43 127)(44 128)(45 121)(46 122)(47 123)(48 124)(49 133)(50 134)(51 135)(52 136)(53 129)(54 130)(55 131)(56 132)(57 115)(58 116)(59 117)(60 118)(61 119)(62 120)(63 113)(64 114)(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 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 97)(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 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 79)(58 80)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(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 42 62 69)(2 51 43 63 70)(3 52 44 64 71)(4 53 45 57 72)(5 54 46 58 65)(6 55 47 59 66)(7 56 48 60 67)(8 49 41 61 68)(9 36 28 20 158)(10 37 29 21 159)(11 38 30 22 160)(12 39 31 23 153)(13 40 32 24 154)(14 33 25 17 155)(15 34 26 18 156)(16 35 27 19 157)(73 106 103 95 87)(74 107 104 96 88)(75 108 97 89 81)(76 109 98 90 82)(77 110 99 91 83)(78 111 100 92 84)(79 112 101 93 85)(80 105 102 94 86)(113 146 143 135 127)(114 147 144 136 128)(115 148 137 129 121)(116 149 138 130 122)(117 150 139 131 123)(118 151 140 132 124)(119 152 141 133 125)(120 145 142 134 126)
(1 109)(2 110)(3 111)(4 112)(5 105)(6 106)(7 107)(8 108)(9 140)(10 141)(11 142)(12 143)(13 144)(14 137)(15 138)(16 139)(17 121)(18 122)(19 123)(20 124)(21 125)(22 126)(23 127)(24 128)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 113)(32 114)(33 148)(34 149)(35 150)(36 151)(37 152)(38 145)(39 146)(40 147)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 73)(56 74)(57 93)(58 94)(59 95)(60 96)(61 89)(62 90)(63 91)(64 92)(65 102)(66 103)(67 104)(68 97)(69 98)(70 99)(71 100)(72 101)(129 155)(130 156)(131 157)(132 158)(133 159)(134 160)(135 153)(136 154)

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

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

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

128 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4H 4I ··· 4P 5A 5B 8A ··· 8P 8Q ··· 8AF 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 ··· 1 5 ··· 5 1 ··· 1 5 ··· 5 2 2 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 D5 D10 D10 C4×D5 C4×D5 C8×D5 kernel D5×C22×C8 D5×C2×C8 C22×C5⋊2C8 C22×C40 D5×C22×C4 C2×C4×D5 C22×Dic5 C23×D5 C22×D5 C22×C8 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 12 1 1 1 12 2 2 32 2 12 2 12 4 32

Matrix representation of D5×C22×C8 in GL4(𝔽41) generated by

 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 1 0 0 40 34
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 34 40
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] >;

D5×C22×C8 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

׿
×
𝔽