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## G = C22×D5⋊C8order 320 = 26·5

### Direct product of C22 and D5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C22×D5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C5⋊C8 — C2×C5⋊C8 — C22×C5⋊C8 — C22×D5⋊C8
 Lower central C5 — C22×D5⋊C8
 Upper central C1 — C22×C4

Generators and relations for C22×D5⋊C8
G = < a,b,c,d,e | a2=b2=c5=d2=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ece-1=c3, ede-1=c2d >

Subgroups: 906 in 338 conjugacy classes, 196 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×4], C4 [×4], C22 [×7], C22 [×28], C5, C8 [×8], C2×C4 [×6], C2×C4 [×22], C23, C23 [×14], D5 [×8], C10, C10 [×6], C2×C8 [×28], C22×C4, C22×C4 [×13], C24, Dic5, Dic5 [×3], C20 [×4], D10 [×28], C2×C10 [×7], C22×C8 [×14], C23×C4, C5⋊C8 [×8], C4×D5 [×16], C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×14], C22×C10, C23×C8, D5⋊C8 [×16], C2×C5⋊C8 [×12], C2×C4×D5 [×12], C22×Dic5, C22×C20, C23×D5, C2×D5⋊C8 [×12], C22×C5⋊C8 [×2], D5×C22×C4, C22×D5⋊C8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, F5, C22×C8 [×14], C23×C4, C2×F5 [×7], C23×C8, D5⋊C8 [×4], C22×F5 [×7], C2×D5⋊C8 [×6], C23×F5, C22×D5⋊C8

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

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

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

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

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 C8 F5 C2×F5 C2×F5 D5⋊C8 kernel C22×D5⋊C8 C2×D5⋊C8 C22×C5⋊C8 D5×C22×C4 C2×C4×D5 C22×C20 C23×D5 C22×D5 C22×C4 C2×C4 C23 C22 # reps 1 12 2 1 12 2 2 32 1 6 1 8

Matrix representation of C22×D5⋊C8 in GL6(𝔽41)

 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 40 0 0 0 1 0 40 0 0 1 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 38 0 0 0 0 0 0 2 39 38 0 0 0 40 39 0 2 0 0 2 0 39 40 0 0 0 38 39 2

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,40,40,40,40],[1,0,0,0,0,0,0,38,0,0,0,0,0,0,2,40,2,0,0,0,39,39,0,38,0,0,38,0,39,39,0,0,0,2,40,2] >;

C22×D5⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xD5:C8");
// GroupNames label

G:=SmallGroup(320,1587);
// by ID

G=gap.SmallGroup(320,1587);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,136,102,6278,818]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^5=d^2=e^8=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,d*c*d=c^-1,e*c*e^-1=c^3,e*d*e^-1=c^2*d>;
// generators/relations

׿
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