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## G = C23×C5⋊D4order 320 = 26·5

### Direct product of C23 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C23×C5⋊D4
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C23×D5 — D5×C24 — C23×C5⋊D4
 Lower central C5 — C10 — C23×C5⋊D4
 Upper central C1 — C24 — C25

Generators and relations for C23×C5⋊D4
G = < a,b,c,d,e,f | a2=b2=c2=d5=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 4126 in 1362 conjugacy classes, 543 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, D4, C23, C23, D5, C10, C10, C10, C22×C4, C2×D4, C24, C24, C24, Dic5, D10, D10, C2×C10, C2×C10, C23×C4, C22×D4, C25, C25, C2×Dic5, C5⋊D4, C22×D5, C22×D5, C22×C10, C22×C10, D4×C23, C22×Dic5, C2×C5⋊D4, C23×D5, C23×D5, C23×C10, C23×C10, C23×C10, C23×Dic5, C22×C5⋊D4, D5×C24, C24×C10, C23×C5⋊D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C25, C5⋊D4, C22×D5, D4×C23, C2×C5⋊D4, C23×D5, C22×C5⋊D4, D5×C24, C23×C5⋊D4

Smallest permutation representation of C23×C5⋊D4
On 160 points
Generators in S160
(1 86)(2 87)(3 88)(4 89)(5 90)(6 81)(7 82)(8 83)(9 84)(10 85)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(81 141)(82 142)(83 143)(84 144)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)(97 157)(98 158)(99 159)(100 160)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(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)
(1 131 6 136)(2 135 7 140)(3 134 8 139)(4 133 9 138)(5 132 10 137)(11 126 16 121)(12 130 17 125)(13 129 18 124)(14 128 19 123)(15 127 20 122)(21 151 26 156)(22 155 27 160)(23 154 28 159)(24 153 29 158)(25 152 30 157)(31 146 36 141)(32 150 37 145)(33 149 38 144)(34 148 39 143)(35 147 40 142)(41 91 46 96)(42 95 47 100)(43 94 48 99)(44 93 49 98)(45 92 50 97)(51 86 56 81)(52 90 57 85)(53 89 58 84)(54 88 59 83)(55 87 60 82)(61 111 66 116)(62 115 67 120)(63 114 68 119)(64 113 69 118)(65 112 70 117)(71 106 76 101)(72 110 77 105)(73 109 78 104)(74 108 79 103)(75 107 80 102)
(2 5)(3 4)(7 10)(8 9)(11 16)(12 20)(13 19)(14 18)(15 17)(22 25)(23 24)(27 30)(28 29)(31 36)(32 40)(33 39)(34 38)(35 37)(42 45)(43 44)(47 50)(48 49)(51 56)(52 60)(53 59)(54 58)(55 57)(62 65)(63 64)(67 70)(68 69)(71 76)(72 80)(73 79)(74 78)(75 77)(82 85)(83 84)(87 90)(88 89)(91 96)(92 100)(93 99)(94 98)(95 97)(102 105)(103 104)(107 110)(108 109)(111 116)(112 120)(113 119)(114 118)(115 117)(122 125)(123 124)(127 130)(128 129)(131 136)(132 140)(133 139)(134 138)(135 137)(142 145)(143 144)(147 150)(148 149)(151 156)(152 160)(153 159)(154 158)(155 157)

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

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

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

104 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2W 2X ··· 2AE 4A ··· 4H 5A 5B 10A ··· 10BJ order 1 2 ··· 2 2 ··· 2 2 ··· 2 4 ··· 4 5 5 10 ··· 10 size 1 1 ··· 1 2 ··· 2 10 ··· 10 10 ··· 10 2 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 D4 D5 D10 C5⋊D4 kernel C23×C5⋊D4 C23×Dic5 C22×C5⋊D4 D5×C24 C24×C10 C22×C10 C25 C24 C23 # reps 1 1 28 1 1 8 2 30 32

Matrix representation of C23×C5⋊D4 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
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 34 40 0 0 0 0 1 0 0 0 0 0 0 0 34 40 0 0 0 0 1 0
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 34 40 0 0 0 0 0 0 24 1 0 0 0 0 38 17
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 34 40 0 0 0 0 0 0 1 0 0 0 0 0 34 40

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],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,34,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,24,38,0,0,0,0,1,17],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40] >;

C23×C5⋊D4 in GAP, Magma, Sage, TeX

C_2^3\times C_5\rtimes D_4
% in TeX

G:=Group("C2^3xC5:D4");
// GroupNames label

G:=SmallGroup(320,1627);
// by ID

G=gap.SmallGroup(320,1627);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,1684,12550]);
// Polycyclic

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

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