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## G = C4×C23.D5order 320 = 26·5

### Direct product of C4 and C23.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4×C23.D5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C23.D5 — C4×C23.D5
 Lower central C5 — C10 — C4×C23.D5
 Upper central C1 — C22×C4 — C23×C4

Generators and relations for C4×C23.D5
G = < a,b,c,d,e,f | a4=b2=c2=d2=e5=1, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 638 in 258 conjugacy classes, 119 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2×C42, C2×C22⋊C4, C23×C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C4×C22⋊C4, C4×Dic5, C23.D5, C22×Dic5, C22×C20, C22×C20, C22×C20, C23×C10, C10.10C42, C2×C4×Dic5, C2×C23.D5, C23×C20, C4×C23.D5
Quotients:

Smallest permutation representation of C4×C23.D5
On 160 points
Generators in S160
(1 51 11 41)(2 52 12 42)(3 53 13 43)(4 54 14 44)(5 55 15 45)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 131 91 121)(82 132 92 122)(83 133 93 123)(84 134 94 124)(85 135 95 125)(86 136 96 126)(87 137 97 127)(88 138 98 128)(89 139 99 129)(90 140 100 130)(101 151 111 141)(102 152 112 142)(103 153 113 143)(104 154 114 144)(105 155 115 145)(106 156 116 146)(107 157 117 147)(108 158 118 148)(109 159 119 149)(110 160 120 150)
(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)(81 106)(82 107)(83 108)(84 109)(85 110)(86 101)(87 102)(88 103)(89 104)(90 105)(91 116)(92 117)(93 118)(94 119)(95 120)(96 111)(97 112)(98 113)(99 114)(100 115)(121 146)(122 147)(123 148)(124 149)(125 150)(126 141)(127 142)(128 143)(129 144)(130 145)(131 156)(132 157)(133 158)(134 159)(135 160)(136 151)(137 152)(138 153)(139 154)(140 155)
(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 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)(81 86)(82 87)(83 88)(84 89)(85 90)(91 96)(92 97)(93 98)(94 99)(95 100)(101 106)(102 107)(103 108)(104 109)(105 110)(111 116)(112 117)(113 118)(114 119)(115 120)(121 126)(122 127)(123 128)(124 129)(125 130)(131 136)(132 137)(133 138)(134 139)(135 140)(141 146)(142 147)(143 148)(144 149)(145 150)(151 156)(152 157)(153 158)(154 159)(155 160)
(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)
(1 103 21 83)(2 102 22 82)(3 101 23 81)(4 105 24 85)(5 104 25 84)(6 108 26 88)(7 107 27 87)(8 106 28 86)(9 110 29 90)(10 109 30 89)(11 113 31 93)(12 112 32 92)(13 111 33 91)(14 115 34 95)(15 114 35 94)(16 118 36 98)(17 117 37 97)(18 116 38 96)(19 120 39 100)(20 119 40 99)(41 143 61 123)(42 142 62 122)(43 141 63 121)(44 145 64 125)(45 144 65 124)(46 148 66 128)(47 147 67 127)(48 146 68 126)(49 150 69 130)(50 149 70 129)(51 153 71 133)(52 152 72 132)(53 151 73 131)(54 155 74 135)(55 154 75 134)(56 158 76 138)(57 157 77 137)(58 156 78 136)(59 160 79 140)(60 159 80 139)

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

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

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

104 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I 4J 4K 4L 4M ··· 4AB 5A 5B 10A ··· 10AD 20A ··· 20AF order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + image C1 C2 C2 C2 C2 C4 C4 D4 D5 C4○D4 Dic5 D10 D10 C5⋊D4 C4×D5 C4○D20 kernel C4×C23.D5 C10.10C42 C2×C4×Dic5 C2×C23.D5 C23×C20 C23.D5 C22×C20 C2×C20 C23×C4 C2×C10 C22×C4 C22×C4 C24 C2×C4 C23 C22 # reps 1 2 2 2 1 16 8 4 2 4 8 4 2 16 16 16

Matrix representation of C4×C23.D5 in GL4(𝔽41) generated by

 9 0 0 0 0 32 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 1
,
 40 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 10 0 0 0 0 37
,
 9 0 0 0 0 40 0 0 0 0 0 37 0 0 31 0
G:=sub<GL(4,GF(41))| [9,0,0,0,0,32,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,37],[9,0,0,0,0,40,0,0,0,0,0,31,0,0,37,0] >;

C4×C23.D5 in GAP, Magma, Sage, TeX

C_4\times C_2^3.D_5
% in TeX

G:=Group("C4xC2^3.D5");
// GroupNames label

G:=SmallGroup(320,836);
// by ID

G=gap.SmallGroup(320,836);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,12550]);
// Polycyclic

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

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