direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C23.36D4, C4○D4⋊2C20, D4⋊4(C2×C20), Q8⋊4(C2×C20), C4.54(D4×C10), (C2×C20).314D4, C20.461(C2×D4), C4.4(C22×C20), D4⋊C4⋊14C10, C23.36(C5×D4), Q8⋊C4⋊14C10, C22.44(D4×C10), (C10×M4(2))⋊28C2, (C2×M4(2))⋊10C10, (C2×C40).322C22, (C2×C20).893C23, C20.208(C22×C4), (C22×C10).158D4, C20.155(C22⋊C4), C10.126(C8⋊C22), (D4×C10).288C22, (Q8×C10).252C22, C10.126(C8.C22), (C22×C20).410C22, (C2×C4⋊C4)⋊10C10, (C10×C4⋊C4)⋊37C2, (C5×C4○D4)⋊14C4, (C5×D4)⋊34(C2×C4), (C5×Q8)⋊31(C2×C4), C2.1(C5×C8⋊C22), C4⋊C4.39(C2×C10), (C2×C4).20(C2×C20), (C2×C8).47(C2×C10), (C2×C4○D4).5C10, (C2×C4).123(C5×D4), C4.23(C5×C22⋊C4), (C5×D4⋊C4)⋊37C2, C2.1(C5×C8.C22), (C2×C20).366(C2×C4), (C5×Q8⋊C4)⋊37C2, (C10×C4○D4).19C2, (C2×D4).46(C2×C10), (C2×C10).620(C2×D4), C2.20(C10×C22⋊C4), (C2×Q8).37(C2×C10), C22.5(C5×C22⋊C4), (C5×C4⋊C4).360C22, C10.149(C2×C22⋊C4), (C22×C4).29(C2×C10), (C2×C4).68(C22×C10), (C2×C10).96(C22⋊C4), SmallGroup(320,918)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.36D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >
Subgroups: 274 in 162 conjugacy classes, 82 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C20, C20, C20, C2×C10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, C23.36D4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×M4(2), C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C5×D4⋊C4, C5×Q8⋊C4, C10×C4⋊C4, C10×M4(2), C10×C4○D4, C5×C23.36D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C22⋊C4, C8⋊C22, C8.C22, C2×C20, C5×D4, C22×C10, C23.36D4, C5×C22⋊C4, C22×C20, D4×C10, C10×C22⋊C4, C5×C8⋊C22, C5×C8.C22, C5×C23.36D4
►On 160 points
Generators in S160
(1 98 23 90 15)(2 99 24 91 16)(3 100 17 92 9)(4 101 18 93 10)(5 102 19 94 11)(6 103 20 95 12)(7 104 21 96 13)(8 97 22 89 14)(25 42 116 33 108)(26 43 117 34 109)(27 44 118 35 110)(28 45 119 36 111)(29 46 120 37 112)(30 47 113 38 105)(31 48 114 39 106)(32 41 115 40 107)(49 123 139 57 131)(50 124 140 58 132)(51 125 141 59 133)(52 126 142 60 134)(53 127 143 61 135)(54 128 144 62 136)(55 121 137 63 129)(56 122 138 64 130)(65 81 155 73 147)(66 82 156 74 148)(67 83 157 75 149)(68 84 158 76 150)(69 85 159 77 151)(70 86 160 78 152)(71 87 153 79 145)(72 88 154 80 146)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 108)(10 105)(11 110)(12 107)(13 112)(14 109)(15 106)(16 111)(17 116)(18 113)(19 118)(20 115)(21 120)(22 117)(23 114)(24 119)(33 92)(34 89)(35 94)(36 91)(37 96)(38 93)(39 90)(40 95)(41 103)(42 100)(43 97)(44 102)(45 99)(46 104)(47 101)(48 98)(49 151)(50 148)(51 145)(52 150)(53 147)(54 152)(55 149)(56 146)(57 159)(58 156)(59 153)(60 158)(61 155)(62 160)(63 157)(64 154)(65 127)(66 124)(67 121)(68 126)(69 123)(70 128)(71 125)(72 122)(73 135)(74 132)(75 129)(76 134)(77 131)(78 136)(79 133)(80 130)(81 143)(82 140)(83 137)(84 142)(85 139)(86 144)(87 141)(88 138)
(1 127)(2 128)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 65)(32 66)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 73)(40 74)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 81)(89 134)(90 135)(91 136)(92 129)(93 130)(94 131)(95 132)(96 133)(97 142)(98 143)(99 144)(100 137)(101 138)(102 139)(103 140)(104 141)(105 146)(106 147)(107 148)(108 149)(109 150)(110 151)(111 152)(112 145)(113 154)(114 155)(115 156)(116 157)(117 158)(118 159)(119 160)(120 153)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(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 8 127 126)(2 125 128 7)(3 6 121 124)(4 123 122 5)(9 12 55 50)(10 49 56 11)(13 16 51 54)(14 53 52 15)(17 20 63 58)(18 57 64 19)(21 24 59 62)(22 61 60 23)(25 32 67 66)(26 65 68 31)(27 30 69 72)(28 71 70 29)(33 40 75 74)(34 73 76 39)(35 38 77 80)(36 79 78 37)(41 83 82 42)(43 81 84 48)(44 47 85 88)(45 87 86 46)(89 135 134 90)(91 133 136 96)(92 95 129 132)(93 131 130 94)(97 143 142 98)(99 141 144 104)(100 103 137 140)(101 139 138 102)(105 151 146 110)(106 109 147 150)(107 149 148 108)(111 145 152 112)(113 159 154 118)(114 117 155 158)(115 157 156 116)(119 153 160 120)
G:=sub<Sym(160)| (1,98,23,90,15)(2,99,24,91,16)(3,100,17,92,9)(4,101,18,93,10)(5,102,19,94,11)(6,103,20,95,12)(7,104,21,96,13)(8,97,22,89,14)(25,42,116,33,108)(26,43,117,34,109)(27,44,118,35,110)(28,45,119,36,111)(29,46,120,37,112)(30,47,113,38,105)(31,48,114,39,106)(32,41,115,40,107)(49,123,139,57,131)(50,124,140,58,132)(51,125,141,59,133)(52,126,142,60,134)(53,127,143,61,135)(54,128,144,62,136)(55,121,137,63,129)(56,122,138,64,130)(65,81,155,73,147)(66,82,156,74,148)(67,83,157,75,149)(68,84,158,76,150)(69,85,159,77,151)(70,86,160,78,152)(71,87,153,79,145)(72,88,154,80,146), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,108)(10,105)(11,110)(12,107)(13,112)(14,109)(15,106)(16,111)(17,116)(18,113)(19,118)(20,115)(21,120)(22,117)(23,114)(24,119)(33,92)(34,89)(35,94)(36,91)(37,96)(38,93)(39,90)(40,95)(41,103)(42,100)(43,97)(44,102)(45,99)(46,104)(47,101)(48,98)(49,151)(50,148)(51,145)(52,150)(53,147)(54,152)(55,149)(56,146)(57,159)(58,156)(59,153)(60,158)(61,155)(62,160)(63,157)(64,154)(65,127)(66,124)(67,121)(68,126)(69,123)(70,128)(71,125)(72,122)(73,135)(74,132)(75,129)(76,134)(77,131)(78,136)(79,133)(80,130)(81,143)(82,140)(83,137)(84,142)(85,139)(86,144)(87,141)(88,138), (1,127)(2,128)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,65)(32,66)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,81)(89,134)(90,135)(91,136)(92,129)(93,130)(94,131)(95,132)(96,133)(97,142)(98,143)(99,144)(100,137)(101,138)(102,139)(103,140)(104,141)(105,146)(106,147)(107,148)(108,149)(109,150)(110,151)(111,152)(112,145)(113,154)(114,155)(115,156)(116,157)(117,158)(118,159)(119,160)(120,153), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (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), (1,8,127,126)(2,125,128,7)(3,6,121,124)(4,123,122,5)(9,12,55,50)(10,49,56,11)(13,16,51,54)(14,53,52,15)(17,20,63,58)(18,57,64,19)(21,24,59,62)(22,61,60,23)(25,32,67,66)(26,65,68,31)(27,30,69,72)(28,71,70,29)(33,40,75,74)(34,73,76,39)(35,38,77,80)(36,79,78,37)(41,83,82,42)(43,81,84,48)(44,47,85,88)(45,87,86,46)(89,135,134,90)(91,133,136,96)(92,95,129,132)(93,131,130,94)(97,143,142,98)(99,141,144,104)(100,103,137,140)(101,139,138,102)(105,151,146,110)(106,109,147,150)(107,149,148,108)(111,145,152,112)(113,159,154,118)(114,117,155,158)(115,157,156,116)(119,153,160,120)>;
G:=Group( (1,98,23,90,15)(2,99,24,91,16)(3,100,17,92,9)(4,101,18,93,10)(5,102,19,94,11)(6,103,20,95,12)(7,104,21,96,13)(8,97,22,89,14)(25,42,116,33,108)(26,43,117,34,109)(27,44,118,35,110)(28,45,119,36,111)(29,46,120,37,112)(30,47,113,38,105)(31,48,114,39,106)(32,41,115,40,107)(49,123,139,57,131)(50,124,140,58,132)(51,125,141,59,133)(52,126,142,60,134)(53,127,143,61,135)(54,128,144,62,136)(55,121,137,63,129)(56,122,138,64,130)(65,81,155,73,147)(66,82,156,74,148)(67,83,157,75,149)(68,84,158,76,150)(69,85,159,77,151)(70,86,160,78,152)(71,87,153,79,145)(72,88,154,80,146), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,108)(10,105)(11,110)(12,107)(13,112)(14,109)(15,106)(16,111)(17,116)(18,113)(19,118)(20,115)(21,120)(22,117)(23,114)(24,119)(33,92)(34,89)(35,94)(36,91)(37,96)(38,93)(39,90)(40,95)(41,103)(42,100)(43,97)(44,102)(45,99)(46,104)(47,101)(48,98)(49,151)(50,148)(51,145)(52,150)(53,147)(54,152)(55,149)(56,146)(57,159)(58,156)(59,153)(60,158)(61,155)(62,160)(63,157)(64,154)(65,127)(66,124)(67,121)(68,126)(69,123)(70,128)(71,125)(72,122)(73,135)(74,132)(75,129)(76,134)(77,131)(78,136)(79,133)(80,130)(81,143)(82,140)(83,137)(84,142)(85,139)(86,144)(87,141)(88,138), (1,127)(2,128)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,65)(32,66)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,81)(89,134)(90,135)(91,136)(92,129)(93,130)(94,131)(95,132)(96,133)(97,142)(98,143)(99,144)(100,137)(101,138)(102,139)(103,140)(104,141)(105,146)(106,147)(107,148)(108,149)(109,150)(110,151)(111,152)(112,145)(113,154)(114,155)(115,156)(116,157)(117,158)(118,159)(119,160)(120,153), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (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), (1,8,127,126)(2,125,128,7)(3,6,121,124)(4,123,122,5)(9,12,55,50)(10,49,56,11)(13,16,51,54)(14,53,52,15)(17,20,63,58)(18,57,64,19)(21,24,59,62)(22,61,60,23)(25,32,67,66)(26,65,68,31)(27,30,69,72)(28,71,70,29)(33,40,75,74)(34,73,76,39)(35,38,77,80)(36,79,78,37)(41,83,82,42)(43,81,84,48)(44,47,85,88)(45,87,86,46)(89,135,134,90)(91,133,136,96)(92,95,129,132)(93,131,130,94)(97,143,142,98)(99,141,144,104)(100,103,137,140)(101,139,138,102)(105,151,146,110)(106,109,147,150)(107,149,148,108)(111,145,152,112)(113,159,154,118)(114,117,155,158)(115,157,156,116)(119,153,160,120) );
G=PermutationGroup([[(1,98,23,90,15),(2,99,24,91,16),(3,100,17,92,9),(4,101,18,93,10),(5,102,19,94,11),(6,103,20,95,12),(7,104,21,96,13),(8,97,22,89,14),(25,42,116,33,108),(26,43,117,34,109),(27,44,118,35,110),(28,45,119,36,111),(29,46,120,37,112),(30,47,113,38,105),(31,48,114,39,106),(32,41,115,40,107),(49,123,139,57,131),(50,124,140,58,132),(51,125,141,59,133),(52,126,142,60,134),(53,127,143,61,135),(54,128,144,62,136),(55,121,137,63,129),(56,122,138,64,130),(65,81,155,73,147),(66,82,156,74,148),(67,83,157,75,149),(68,84,158,76,150),(69,85,159,77,151),(70,86,160,78,152),(71,87,153,79,145),(72,88,154,80,146)], [(1,31),(2,28),(3,25),(4,30),(5,27),(6,32),(7,29),(8,26),(9,108),(10,105),(11,110),(12,107),(13,112),(14,109),(15,106),(16,111),(17,116),(18,113),(19,118),(20,115),(21,120),(22,117),(23,114),(24,119),(33,92),(34,89),(35,94),(36,91),(37,96),(38,93),(39,90),(40,95),(41,103),(42,100),(43,97),(44,102),(45,99),(46,104),(47,101),(48,98),(49,151),(50,148),(51,145),(52,150),(53,147),(54,152),(55,149),(56,146),(57,159),(58,156),(59,153),(60,158),(61,155),(62,160),(63,157),(64,154),(65,127),(66,124),(67,121),(68,126),(69,123),(70,128),(71,125),(72,122),(73,135),(74,132),(75,129),(76,134),(77,131),(78,136),(79,133),(80,130),(81,143),(82,140),(83,137),(84,142),(85,139),(86,144),(87,141),(88,138)], [(1,127),(2,128),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,55),(10,56),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,63),(18,64),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,65),(32,66),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,73),(40,74),(41,82),(42,83),(43,84),(44,85),(45,86),(46,87),(47,88),(48,81),(89,134),(90,135),(91,136),(92,129),(93,130),(94,131),(95,132),(96,133),(97,142),(98,143),(99,144),(100,137),(101,138),(102,139),(103,140),(104,141),(105,146),(106,147),(107,148),(108,149),(109,150),(110,151),(111,152),(112,145),(113,154),(114,155),(115,156),(116,157),(117,158),(118,159),(119,160),(120,153)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160)], [(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)], [(1,8,127,126),(2,125,128,7),(3,6,121,124),(4,123,122,5),(9,12,55,50),(10,49,56,11),(13,16,51,54),(14,53,52,15),(17,20,63,58),(18,57,64,19),(21,24,59,62),(22,61,60,23),(25,32,67,66),(26,65,68,31),(27,30,69,72),(28,71,70,29),(33,40,75,74),(34,73,76,39),(35,38,77,80),(36,79,78,37),(41,83,82,42),(43,81,84,48),(44,47,85,88),(45,87,86,46),(89,135,134,90),(91,133,136,96),(92,95,129,132),(93,131,130,94),(97,143,142,98),(99,141,144,104),(100,103,137,140),(101,139,138,102),(105,151,146,110),(106,109,147,150),(107,149,148,108),(111,145,152,112),(113,159,154,118),(114,117,155,158),(115,157,156,116),(119,153,160,120)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20AN | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | D4 | D4 | C5×D4 | C5×D4 | C8⋊C22 | C8.C22 | C5×C8⋊C22 | C5×C8.C22 |
| kernel | C5×C23.36D4 | C5×D4⋊C4 | C5×Q8⋊C4 | C10×C4⋊C4 | C10×M4(2) | C10×C4○D4 | C5×C4○D4 | C23.36D4 | D4⋊C4 | Q8⋊C4 | C2×C4⋊C4 | C2×M4(2) | C2×C4○D4 | C4○D4 | C2×C20 | C22×C10 | C2×C4 | C23 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 8 | 8 | 4 | 4 | 4 | 32 | 3 | 1 | 12 | 4 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C23.36D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 24 | 39 | 0 | 0 | 0 | 0 |
| 21 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 39 | 4 | 4 |
| 0 | 0 | 2 | 2 | 37 | 4 |
| 0 | 0 | 37 | 37 | 39 | 2 |
| 0 | 0 | 4 | 37 | 39 | 39 |
| 17 | 2 | 0 | 0 | 0 | 0 |
| 19 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 2 | 37 | 37 |
| 0 | 0 | 2 | 2 | 37 | 4 |
| 0 | 0 | 37 | 37 | 39 | 2 |
| 0 | 0 | 37 | 4 | 2 | 2 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[24,21,0,0,0,0,39,17,0,0,0,0,0,0,2,2,37,4,0,0,39,2,37,37,0,0,4,37,39,39,0,0,4,4,2,39],[17,19,0,0,0,0,2,24,0,0,0,0,0,0,39,2,37,37,0,0,2,2,37,4,0,0,37,37,39,2,0,0,37,4,2,2] >;
C5×C23.36D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{36}D_4 % in TeX
G:=Group("C5xC2^3.36D4"); // GroupNames label
G:=SmallGroup(320,918);
// by ID
G=gap.SmallGroup(320,918);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1731,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^4=d,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,e*b*e^-1=b*d=d*b,b*f=f*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=c*d*e^3>;
// generators/relations