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G = C2×C23.11D10  order 320 = 26·5

Direct product of C2 and C23.11D10

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

Aliases: C2×C23.11D10, C24.50D10, C23.57(C4×D5), C10.27(C23×C4), (C2×C10).24C24, C10⋊3(C42⋊C2), (C2×C20).569C23, C22⋊C4.122D10, (C22×Dic5)⋊14C4, (C4×Dic5)⋊71C22, (C22×C4).311D10, (C23×Dic5).7C2, C22.16(C23×D5), C10.D4⋊56C22, (C23×C10).50C22, Dic5.52(C22×C4), C23.143(C22×D5), C23.D5.82C22, C22.63(D4⋊2D5), (C22×C10).386C23, (C22×C20).349C22, (C2×Dic5).369C23, (C22×Dic5).288C22, C2.8(D5×C22×C4), C5⋊3(C2×C42⋊C2), (C2×C4×Dic5)⋊28C2, C22.23(C2×C4×D5), C10.65(C2×C4○D4), C2.1(C2×D4⋊2D5), (C2×Dic5)⋊30(C2×C4), (C2×C22⋊C4).20D5, (C2×C10.D4)⋊33C2, (C10×C22⋊C4).25C2, (C2×C4).254(C22×D5), (C2×C23.D5).19C2, (C2×C10).165(C4○D4), (C22×C10).144(C2×C4), (C2×C10).118(C22×C4), (C5×C22⋊C4).132C22, SmallGroup(320,1152)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C2×C23.11D10
Chief series
C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C2×C23.11D10
Lower central
C5 — C10 — C2×C23.11D10

Subgroups: 830 in 330 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×16], C22, C22 [×10], C22 [×12], C5, C2×C4 [×4], C2×C4 [×40], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C42 [×8], C22⋊C4 [×4], C22⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×16], C24, Dic5 [×8], Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C42 [×2], C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×8], C23×C4, C2×Dic5 [×32], C2×Dic5 [×4], C2×C20 [×4], C2×C20 [×4], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C42⋊C2, C4×Dic5 [×8], C10.D4 [×8], C23.D5 [×4], C5×C22⋊C4 [×4], C22×Dic5 [×2], C22×Dic5 [×14], C22×C20 [×2], C23×C10, C23.11D10 [×8], C2×C4×Dic5 [×2], C2×C10.D4 [×2], C2×C23.D5, C10×C22⋊C4, C23×Dic5, C2×C23.11D10

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C4○D4 [×4], C24, D10 [×7], C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C4×D5 [×4], C22×D5 [×7], C2×C42⋊C2, C2×C4×D5 [×6], D4⋊2D5 [×4], C23×D5, C23.11D10 [×4], D5×C22×C4, C2×D4⋊2D5 [×2], C2×C23.11D10

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, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

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

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

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

(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)

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

Matrix representation ►G ⊆ GL5(𝔽41)

400000
01000
00100
00010
00001
,
10000
040000
004000
00010
000140
,
10000
040000
004000
00010
00001
,
10000
01000
00100
000400
000040
,
10000
001900
0132200
000923
000932
,
10000
032000
0203800
000402
000401

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],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,1,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,13,0,0,0,19,22,0,0,0,0,0,9,9,0,0,0,23,32],[1,0,0,0,0,0,3,20,0,0,0,20,38,0,0,0,0,0,40,40,0,0,0,2,1] >;
 

80 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P4Q···4AB5A5B10A···10N10O···10V20A···20P
order12···222224···44···44···45510···1010···1020···20
size11···122222···25···510···10222···24···44···4

80 irreducible representations

dim111111112222224
type+++++++++++-
imageC1C2C2C2C2C2C2C4D5C4○D4D10D10D10C4×D5D4⋊2D5
kernelC2×C23.11D10C23.11D10C2×C4×Dic5C2×C10.D4C2×C23.D5C10×C22⋊C4C23×Dic5C22×Dic5C2×C22⋊C4C2×C10C22⋊C4C22×C4C24C23C22
# reps18221111628842168

In GAP, Magma, Sage, TeX 

C_2\times C_2^3._{11}D_{10}
 
% in TeX
 
G:=Group("C2xC2^3.11D10");
 
// GroupNames label
 
G:=SmallGroup(320,1152);
 
// by ID
 
G=gap.SmallGroup(320,1152);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,297,80,12550]);
 
// 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,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=e^9>;
 
// generators/relations
 

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×
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ℤ
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