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G = C2×C23.18D10  order 320 = 26·5

Direct product of C2 and C23.18D10

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

Aliases: C2×C23.18D10, C24.58D10, (C2×D4).228D10, (C23×Dic5)⋊8C2, (C22×D4).10D5, (C2×C10).291C24, (C2×C20).642C23, (C22×C10).121D4, (C22×C4).269D10, C10.139(C22×D4), C23.67(C5⋊D4), C23.D5⋊57C22, (D4×C10).311C22, C10.D4⋊72C22, C10⋊5(C22.D4), (C23×C10).73C22, C23.133(C22×D5), C22.305(C23×D5), C22.77(D4⋊2D5), (C22×C10).227C23, (C22×C20).437C22, (C2×Dic5).291C23, (C22×Dic5)⋊48C22, (D4×C2×C10).21C2, (C2×C10).73(C2×D4), C5⋊6(C2×C22.D4), C10.103(C2×C4○D4), C2.67(C2×D4⋊2D5), (C2×C23.D5)⋊24C2, C2.12(C22×C5⋊D4), (C2×C10.D4)⋊47C2, (C2×C4).236(C22×D5), C22.108(C2×C5⋊D4), (C2×C10).175(C4○D4), SmallGroup(320,1468)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C2×C23.18D10
Chief series
C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C2×C23.18D10
Lower central
C5 — C2×C10 — C2×C23.18D10

Subgroups: 958 in 342 conjugacy classes, 127 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×10], C22, C22 [×10], C22 [×22], C5, C2×C4 [×2], C2×C4 [×26], D4 [×8], C23, C23 [×8], C23 [×10], C10, C10 [×6], C10 [×6], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4, C22×C4 [×12], C2×D4 [×4], C2×D4 [×4], C24 [×2], Dic5 [×8], C20 [×2], C2×C10, C2×C10 [×10], C2×C10 [×22], C2×C22⋊C4 [×3], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, C2×Dic5 [×8], C2×Dic5 [×16], C2×C20 [×2], C2×C20 [×2], C5×D4 [×8], C22×C10, C22×C10 [×8], C22×C10 [×10], C2×C22.D4, C10.D4 [×8], C23.D5 [×12], C22×Dic5 [×8], C22×Dic5 [×4], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×C10 [×2], C2×C10.D4 [×2], C23.18D10 [×8], C2×C23.D5, C2×C23.D5 [×2], C23×Dic5, D4×C2×C10, C2×C23.18D10

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C5⋊D4 [×4], C22×D5 [×7], C2×C22.D4, D4⋊2D5 [×4], C2×C5⋊D4 [×6], C23×D5, C23.18D10 [×4], C2×D4⋊2D5 [×2], C22×C5⋊D4, C2×C23.18D10

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

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

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

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

(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)

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

Matrix representation ►G ⊆ GL5(𝔽41)

400000
01000
00100
00010
00001
,
400000
01000
00100
000126
000040
,
10000
040000
004000
000400
000040
,
10000
01000
00100
000400
000040
,
400000
010000
011400
00010
0002240
,
400000
0191200
042200
0003212
00079

G:=sub<GL(5,GF(41))| [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],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,26,40],[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],[40,0,0,0,0,0,10,11,0,0,0,0,4,0,0,0,0,0,1,22,0,0,0,0,40],[40,0,0,0,0,0,19,4,0,0,0,12,22,0,0,0,0,0,32,7,0,0,0,12,9] >;
 

68 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C···4J4K4L4M4N5A5B10A···10N10O···10AD20A···20H
order12···2222222444···444445510···1010···1020···20
size11···12222444410···1020202020222···24···44···4

68 irreducible representations

dim11111122222224
type+++++++++++-
imageC1C2C2C2C2C2D4D5C4○D4D10D10D10C5⋊D4D4⋊2D5
kernelC2×C23.18D10C2×C10.D4C23.18D10C2×C23.D5C23×Dic5D4×C2×C10C22×C10C22×D4C2×C10C22×C4C2×D4C24C23C22
# reps128311428284168

In GAP, Magma, Sage, TeX 

C_2\times C_2^3._{18}D_{10}
 
% in TeX
 
G:=Group("C2xC2^3.18D10");
 
// GroupNames label
 
G:=SmallGroup(320,1468);
 
// by ID
 
G=gap.SmallGroup(320,1468);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,675,297,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=1,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,b*c=c*b,e*b*e^-1=f*b*f^-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=c*e^-1>;
 
// generators/relations
 

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