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G = C2×C23.D10  order 320 = 26·5

Direct product of C2 and C23.D10

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

Aliases: C2×C23.D10, C24.22D10, (C2×C10).26C24, C4⋊Dic5⋊50C22, C22⋊C4.85D10, C10⋊2(C42⋊2C2), (C2×C20).126C23, (C4×Dic5)⋊72C22, (C22×C4).168D10, C23.78(C22×D5), C22.68(C23×D5), C22.71(C4○D20), C10.D4⋊47C22, (C23×C10).52C22, C23.D5.84C22, C22.65(D4⋊2D5), (C22×C20).350C22, (C22×C10).118C23, (C2×Dic5).186C23, (C22×Dic5).226C22, C5⋊2(C2×C42⋊2C2), (C2×C4×Dic5)⋊29C2, (C2×C4⋊Dic5)⋊18C2, C2.13(C2×C4○D20), C10.11(C2×C4○D4), C2.8(C2×D4⋊2D5), (C2×C22⋊C4).17D5, (C2×C10.D4)⋊34C2, (C10×C22⋊C4).20C2, (C2×C4).255(C22×D5), (C2×C23.D5).21C2, (C2×C10).100(C4○D4), (C5×C22⋊C4).108C22, SmallGroup(320,1154)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C2×C23.D10
Chief series
C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C4×Dic5 — C2×C23.D10
Lower central
C5 — C2×C10 — C2×C23.D10

Subgroups: 686 in 246 conjugacy classes, 111 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×12], C22, C22 [×6], C22 [×10], C5, C2×C4 [×4], C2×C4 [×20], C23, C23 [×2], C23 [×6], C10 [×3], C10 [×4], C10 [×2], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×4], C24, Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C42⋊2C2 [×8], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×4], C2×C20 [×4], C22×C10, C22×C10 [×2], C22×C10 [×6], C2×C42⋊2C2, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×4], C23.D5 [×8], C5×C22⋊C4 [×4], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C23.D10 [×8], C2×C4×Dic5, C2×C10.D4 [×2], C2×C4⋊Dic5, C2×C23.D5 [×2], C10×C22⋊C4, C2×C23.D10

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C42⋊2C2 [×4], C2×C4○D4 [×3], C22×D5 [×7], C2×C42⋊2C2, C4○D20 [×2], D4⋊2D5 [×4], C23×D5, C23.D10 [×4], C2×C4○D20, C2×D4⋊2D5 [×2], C2×C23.D10

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

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

(2 152)(4 154)(6 156)(8 158)(10 160)(12 142)(14 144)(16 146)(18 148)(20 150)(21 31)(22 72)(23 33)(24 74)(25 35)(26 76)(27 37)(28 78)(29 39)(30 80)(32 62)(34 64)(36 66)(38 68)(40 70)(41 107)(42 52)(43 109)(44 54)(45 111)(46 56)(47 113)(48 58)(49 115)(50 60)(51 117)(53 119)(55 101)(57 103)(59 105)(61 71)(63 73)(65 75)(67 77)(69 79)(82 131)(84 133)(86 135)(88 137)(90 139)(92 121)(94 123)(96 125)(98 127)(100 129)(102 112)(104 114)(106 116)(108 118)(110 120)

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

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

(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)

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

Matrix representation ►G ⊆ GL5(𝔽41)

400000
01000
00100
000400
000040
,
400000
01000
004000
00010
0003940
,
10000
040000
004000
000400
000040
,
10000
01000
00100
000400
000040
,
10000
02000
002000
0003232
00009
,
10000
002000
02000
00011
000040

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,39,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,2,0,0,0,0,0,20,0,0,0,0,0,32,0,0,0,0,32,9],[1,0,0,0,0,0,0,2,0,0,0,20,0,0,0,0,0,0,1,0,0,0,0,1,40] >;
 

68 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N4O4P4Q4R5A5B10A···10N10O···10V20A···20P
order12···2224444444···444445510···1010···1020···20
size11···14422224410···1020202020222···24···44···4

68 irreducible representations

dim11111112222224
type+++++++++++-
imageC1C2C2C2C2C2C2D5C4○D4D10D10D10C4○D20D4⋊2D5
kernelC2×C23.D10C23.D10C2×C4×Dic5C2×C10.D4C2×C4⋊Dic5C2×C23.D5C10×C22⋊C4C2×C22⋊C4C2×C10C22⋊C4C22×C4C24C22C22
# reps1812121212842168

In GAP, Magma, Sage, TeX 

C_2\times C_2^3.D_{10}
 
% in TeX
 
G:=Group("C2xC2^3.D10");
 
// GroupNames label
 
G:=SmallGroup(320,1154);
 
// by ID
 
G=gap.SmallGroup(320,1154);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,1571,297,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=c,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^9>;
 
// generators/relations
 

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