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G = C22×C8⋊D5order 320 = 26·5

Direct product of C22 and C8⋊D5

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

Aliases: C22×C8⋊D5, C4012C23, C20.66C24, (C2×C8)⋊36D10, C810(C22×D5), (C22×C8)⋊12D5, (C2×C40)⋊48C22, (C22×C40)⋊18C2, C52C811C23, C104(C2×M4(2)), C23.66(C4×D5), C4.65(C23×D5), C54(C22×M4(2)), (C2×C10)⋊12M4(2), C10.50(C23×C4), (C23×D5).11C4, (C4×D5).69C23, C20.204(C22×C4), (C2×C20).879C23, D10.42(C22×C4), (C22×C4).470D10, (C22×Dic5).23C4, Dic5.42(C22×C4), (C22×C20).567C22, (C2×C4×D5).35C4, C4.120(C2×C4×D5), C22.75(C2×C4×D5), C2.30(D5×C22×C4), (C4×D5).83(C2×C4), (C2×C4).187(C4×D5), (D5×C22×C4).27C2, (C2×C20).430(C2×C4), (C2×C52C8)⋊46C22, (C22×C52C8)⋊22C2, (C2×C4×D5).384C22, (C2×C4).823(C22×D5), (C2×C10).256(C22×C4), (C22×C10).170(C2×C4), (C2×Dic5).159(C2×C4), (C22×D5).111(C2×C4), SmallGroup(320,1409)

Series: Derived Chief Lower central Upper central

C1C10 — C22×C8⋊D5
C1C5C10C20C4×D5C2×C4×D5D5×C22×C4 — C22×C8⋊D5
C5C10 — C22×C8⋊D5
C1C22×C4C22×C8

Generators and relations for C22×C8⋊D5
 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, ece=c5, ede=d-1 >

Subgroups: 862 in 298 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×4], C4, C4 [×3], C4 [×4], C22 [×7], C22 [×16], C5, C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×22], C23, C23 [×10], D5 [×4], C10, C10 [×6], C2×C8 [×6], C2×C8 [×6], M4(2) [×16], C22×C4, C22×C4 [×13], C24, Dic5 [×4], C20, C20 [×3], D10 [×4], D10 [×12], C2×C10 [×7], C22×C8, C22×C8, C2×M4(2) [×12], C23×C4, C52C8 [×4], C40 [×4], C4×D5 [×16], C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×6], C22×D5 [×4], C22×C10, C22×M4(2), C8⋊D5 [×16], C2×C52C8 [×6], C2×C40 [×6], C2×C4×D5 [×12], C22×Dic5, C22×C20, C23×D5, C2×C8⋊D5 [×12], C22×C52C8, C22×C40, D5×C22×C4, C22×C8⋊D5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, M4(2) [×4], C22×C4 [×14], C24, D10 [×7], C2×M4(2) [×6], C23×C4, C4×D5 [×4], C22×D5 [×7], C22×M4(2), C8⋊D5 [×4], C2×C4×D5 [×6], C23×D5, C2×C8⋊D5 [×6], D5×C22×C4, C22×C8⋊D5

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

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

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

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

104 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I4J4K4L5A5B8A···8H8I···8P10A···10N20A···20P40A···40AF
order12···222224···44444558···88···810···1020···2040···40
size11···1101010101···110101010222···210···102···22···22···2

104 irreducible representations

dim111111112222222
type++++++++
imageC1C2C2C2C2C4C4C4D5M4(2)D10D10C4×D5C4×D5C8⋊D5
kernelC22×C8⋊D5C2×C8⋊D5C22×C52C8C22×C40D5×C22×C4C2×C4×D5C22×Dic5C23×D5C22×C8C2×C10C2×C8C22×C4C2×C4C23C22
# reps11211112222812212432

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

40000
0100
0010
0001
,
40000
04000
0010
0001
,
32000
04000
00814
002733
,
1000
0100
003440
0010
,
1000
0100
003440
0077
G:=sub<GL(4,GF(41))| [40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[32,0,0,0,0,40,0,0,0,0,8,27,0,0,14,33],[1,0,0,0,0,1,0,0,0,0,34,1,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,34,7,0,0,40,7] >;

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(320,1409);
// by ID

G=gap.SmallGroup(320,1409);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,1123,80,102,12550]);
// 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,e*c*e=c^5,e*d*e=d^-1>;
// generators/relations

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