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## G = (C22×C8)⋊D5order 320 = 26·5

### 3rd semidirect product of C22×C8 and D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C22×C8)⋊D5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — C2×C4○D20 — (C22×C8)⋊D5
 Lower central C5 — C2×C10 — (C22×C8)⋊D5
 Upper central C1 — C2×C4 — C22×C8

Generators and relations for (C22×C8)⋊D5
G = < a,b,c,d,e | a2=b2=c8=d5=e2=1, ab=ba, ece=ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bc4, cd=dc, ede=d-1 >

Subgroups: 574 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×8], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8 [×4], C22×C8, C2×M4(2), C2×C4○D4, C52C8 [×2], C40 [×2], Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×C10, (C22×C8)⋊C2, C2×C52C8 [×2], C4.Dic5 [×2], C2×C40 [×2], C2×C40 [×2], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, D101C8 [×4], C2×C4.Dic5, C22×C40, C2×C4○D20, (C22×C8)⋊D5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C8○D4 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, (C22×C8)⋊C2, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, D20.3C4 [×2], C2×D10⋊C4, (C22×C8)⋊D5

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

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A ··· 8H 8I 8J 8K 8L 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 ··· 8 8 8 8 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 20 20 1 1 1 1 2 2 20 20 2 2 2 ··· 2 20 20 20 20 2 ··· 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D5 D10 D10 C8○D4 C4×D5 D20 C5⋊D4 C4×D5 D20.3C4 kernel (C22×C8)⋊D5 D10⋊1C8 C2×C4.Dic5 C22×C40 C2×C4○D20 C2×Dic10 C2×D20 C2×C5⋊D4 C2×C20 C22×C8 C2×C8 C22×C4 C10 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 2 2 4 4 2 4 2 8 4 8 8 4 32

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

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 23 6 0 0 35 18 0 0 0 0 18 6 0 0 35 23
,
 28 18 0 0 23 13 0 0 0 0 14 0 0 0 0 14
,
 0 1 0 0 40 6 0 0 0 0 6 1 0 0 40 0
,
 0 40 0 0 40 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[23,35,0,0,6,18,0,0,0,0,18,35,0,0,6,23],[28,23,0,0,18,13,0,0,0,0,14,0,0,0,0,14],[0,40,0,0,1,6,0,0,0,0,6,40,0,0,1,0],[0,40,0,0,40,0,0,0,0,0,0,1,0,0,1,0] >;

(C22×C8)⋊D5 in GAP, Magma, Sage, TeX

(C_2^2\times C_8)\rtimes D_5
% in TeX

G:=Group("(C2^2xC8):D5");
// GroupNames label

G:=SmallGroup(320,737);
// by ID

G=gap.SmallGroup(320,737);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,422,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^8=d^5=e^2=1,a*b=b*a,e*c*e=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b*c^4,c*d=d*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽