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## G = C22×C23.D5order 320 = 26·5

### Direct product of C22 and C23.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C22×C23.D5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C22×C23.D5
 Lower central C5 — C10 — C22×C23.D5
 Upper central C1 — C24 — C25

Generators and relations for C22×C23.D5
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f5=1, g2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, gcg-1=ce=ec, cf=fc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg-1=f-1 >

Subgroups: 1534 in 674 conjugacy classes, 287 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C23, C23, C10, C10, C10, C22⋊C4, C22×C4, C24, C24, C24, Dic5, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C23×C4, C25, C2×Dic5, C2×Dic5, C22×C10, C22×C10, C22×C22⋊C4, C23.D5, C22×Dic5, C22×Dic5, C23×C10, C23×C10, C23×C10, C2×C23.D5, C23×Dic5, C24×C10, C22×C23.D5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, C24, Dic5, D10, C2×C22⋊C4, C23×C4, C22×D4, C2×Dic5, C5⋊D4, C22×D5, C22×C22⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4, C23×D5, C2×C23.D5, C23×Dic5, C22×C5⋊D4, C22×C23.D5

Smallest permutation representation of C22×C23.D5
On 160 points
Generators in S160
(1 36)(2 37)(3 38)(4 39)(5 40)(6 31)(7 32)(8 33)(9 34)(10 35)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)(41 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(51 66)(52 67)(53 68)(54 69)(55 70)(56 61)(57 62)(58 63)(59 64)(60 65)(81 116)(82 117)(83 118)(84 119)(85 120)(86 111)(87 112)(88 113)(89 114)(90 115)(91 106)(92 107)(93 108)(94 109)(95 110)(96 101)(97 102)(98 103)(99 104)(100 105)(121 156)(122 157)(123 158)(124 159)(125 160)(126 151)(127 152)(128 153)(129 154)(130 155)(131 146)(132 147)(133 148)(134 149)(135 150)(136 141)(137 142)(138 143)(139 144)(140 145)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(81 131)(82 132)(83 133)(84 134)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)(101 151)(102 152)(103 153)(104 154)(105 155)(106 156)(107 157)(108 158)(109 159)(110 160)(111 141)(112 142)(113 143)(114 144)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(81 136)(82 137)(83 138)(84 139)(85 140)(86 131)(87 132)(88 133)(89 134)(90 135)(91 126)(92 127)(93 128)(94 129)(95 130)(96 121)(97 122)(98 123)(99 124)(100 125)(101 156)(102 157)(103 158)(104 159)(105 160)(106 151)(107 152)(108 153)(109 154)(110 155)(111 146)(112 147)(113 148)(114 149)(115 150)(116 141)(117 142)(118 143)(119 144)(120 145)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)(81 96)(82 97)(83 98)(84 99)(85 100)(86 91)(87 92)(88 93)(89 94)(90 95)(101 116)(102 117)(103 118)(104 119)(105 120)(106 111)(107 112)(108 113)(109 114)(110 115)(121 136)(122 137)(123 138)(124 139)(125 140)(126 131)(127 132)(128 133)(129 134)(130 135)(141 156)(142 157)(143 158)(144 159)(145 160)(146 151)(147 152)(148 153)(149 154)(150 155)
(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)
(1 118 16 103)(2 117 17 102)(3 116 18 101)(4 120 19 105)(5 119 20 104)(6 113 11 108)(7 112 12 107)(8 111 13 106)(9 115 14 110)(10 114 15 109)(21 98 36 83)(22 97 37 82)(23 96 38 81)(24 100 39 85)(25 99 40 84)(26 93 31 88)(27 92 32 87)(28 91 33 86)(29 95 34 90)(30 94 35 89)(41 158 56 143)(42 157 57 142)(43 156 58 141)(44 160 59 145)(45 159 60 144)(46 153 51 148)(47 152 52 147)(48 151 53 146)(49 155 54 150)(50 154 55 149)(61 138 76 123)(62 137 77 122)(63 136 78 121)(64 140 79 125)(65 139 80 124)(66 133 71 128)(67 132 72 127)(68 131 73 126)(69 135 74 130)(70 134 75 129)

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

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

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

104 conjugacy classes

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

104 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 type + + + + + + - + image C1 C2 C2 C2 C4 D4 D5 Dic5 D10 C5⋊D4 kernel C22×C23.D5 C2×C23.D5 C23×Dic5 C24×C10 C23×C10 C22×C10 C25 C24 C24 C23 # reps 1 12 2 1 16 8 2 16 14 32

Matrix representation of C22×C23.D5 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40
,
 40 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 1
,
 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 18
,
 9 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 0 23 0 0 0 25 0

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,18],[9,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,0,25,0,0,0,23,0] >;

C22×C23.D5 in GAP, Magma, Sage, TeX

C_2^2\times C_2^3.D_5
% in TeX

G:=Group("C2^2xC2^3.D5");
// GroupNames label

G:=SmallGroup(320,1511);
// by ID

G=gap.SmallGroup(320,1511);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1123,12550]);
// Polycyclic

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

׿
×
𝔽