direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C24.3C22, (C2×D4)⋊4C20, (C2×C20)⋊29D4, C2.9(D4×C20), (D4×C10)⋊28C4, (C2×C42)⋊5C10, C20⋊9(C22⋊C4), C10.140(C4×D4), C23.9(C2×C20), C24.3(C2×C10), (C22×D4).2C10, C22.39(D4×C10), C10.39(C4⋊1D4), (C23×C10).3C22, C10.136(C4⋊D4), C10.65(C4.4D4), C22.39(C22×C20), C23.63(C22×C10), (C22×C20).576C22, (C22×C10).454C23, (C2×C4×C20)⋊18C2, (C2×C4)⋊6(C5×D4), (C2×C4⋊C4)⋊4C10, C4⋊1(C5×C22⋊C4), (C10×C4⋊C4)⋊31C2, (D4×C2×C10).14C2, C2.5(C5×C4⋊D4), C2.2(C5×C4⋊1D4), (C2×C22⋊C4)⋊3C10, (C10×C22⋊C4)⋊7C2, (C2×C4).44(C2×C20), C2.8(C10×C22⋊C4), C2.3(C5×C4.4D4), (C2×C20).438(C2×C4), (C2×C10).606(C2×D4), C22.24(C5×C4○D4), C10.136(C2×C22⋊C4), (C22×C10).88(C2×C4), (C22×C4).90(C2×C10), (C2×C10).214(C4○D4), (C2×C10).327(C22×C4), SmallGroup(320,891)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C24.3C22
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=e2=1, f2=e, g2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf-1=bc=cb, gbg-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd, ef=fe, eg=ge >
Subgroups: 466 in 258 conjugacy classes, 106 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C24.3C22, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, D4×C10, C23×C10, C2×C4×C20, C10×C22⋊C4, C10×C4⋊C4, D4×C2×C10, C5×C24.3C22
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C4○D4, C20, C2×C10, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C2×C20, C5×D4, C22×C10, C24.3C22, C5×C22⋊C4, C22×C20, D4×C10, C5×C4○D4, C10×C22⋊C4, D4×C20, C5×C4⋊D4, C5×C4.4D4, C5×C4⋊1D4, C5×C24.3C22
►On 160 points
Generators in S160
(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 65)(2 61)(3 62)(4 63)(5 64)(6 133)(7 134)(8 135)(9 131)(10 132)(11 85)(12 81)(13 82)(14 83)(15 84)(16 122)(17 123)(18 124)(19 125)(20 121)(21 129)(22 130)(23 126)(24 127)(25 128)(26 90)(27 86)(28 87)(29 88)(30 89)(31 150)(32 146)(33 147)(34 148)(35 149)(36 110)(37 106)(38 107)(39 108)(40 109)(41 95)(42 91)(43 92)(44 93)(45 94)(46 72)(47 73)(48 74)(49 75)(50 71)(51 58)(52 59)(53 60)(54 56)(55 57)(66 77)(67 78)(68 79)(69 80)(70 76)(96 145)(97 141)(98 142)(99 143)(100 144)(101 137)(102 138)(103 139)(104 140)(105 136)(111 160)(112 156)(113 157)(114 158)(115 159)(116 152)(117 153)(118 154)(119 155)(120 151)
(1 45)(2 41)(3 42)(4 43)(5 44)(6 140)(7 136)(8 137)(9 138)(10 139)(11 53)(12 54)(13 55)(14 51)(15 52)(16 141)(17 142)(18 143)(19 144)(20 145)(21 155)(22 151)(23 152)(24 153)(25 154)(26 69)(27 70)(28 66)(29 67)(30 68)(31 156)(32 157)(33 158)(34 159)(35 160)(36 46)(37 47)(38 48)(39 49)(40 50)(56 81)(57 82)(58 83)(59 84)(60 85)(61 95)(62 91)(63 92)(64 93)(65 94)(71 109)(72 110)(73 106)(74 107)(75 108)(76 86)(77 87)(78 88)(79 89)(80 90)(96 121)(97 122)(98 123)(99 124)(100 125)(101 135)(102 131)(103 132)(104 133)(105 134)(111 149)(112 150)(113 146)(114 147)(115 148)(116 126)(117 127)(118 128)(119 129)(120 130)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 23)(7 24)(8 25)(9 21)(10 22)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 34)(18 35)(19 31)(20 32)(26 43)(27 44)(28 45)(29 41)(30 42)(46 53)(47 54)(48 55)(49 51)(50 52)(56 73)(57 74)(58 75)(59 71)(60 72)(61 78)(62 79)(63 80)(64 76)(65 77)(81 106)(82 107)(83 108)(84 109)(85 110)(86 93)(87 94)(88 95)(89 91)(90 92)(96 113)(97 114)(98 115)(99 111)(100 112)(101 118)(102 119)(103 120)(104 116)(105 117)(121 146)(122 147)(123 148)(124 149)(125 150)(126 133)(127 134)(128 135)(129 131)(130 132)(136 153)(137 154)(138 155)(139 151)(140 152)(141 158)(142 159)(143 160)(144 156)(145 157)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 144)(7 145)(8 141)(9 142)(10 143)(16 137)(17 138)(18 139)(19 140)(20 136)(21 159)(22 160)(23 156)(24 157)(25 158)(26 50)(27 46)(28 47)(29 48)(30 49)(31 152)(32 153)(33 154)(34 155)(35 151)(36 70)(37 66)(38 67)(39 68)(40 69)(41 55)(42 51)(43 52)(44 53)(45 54)(56 94)(57 95)(58 91)(59 92)(60 93)(61 82)(62 83)(63 84)(64 85)(65 81)(71 90)(72 86)(73 87)(74 88)(75 89)(76 110)(77 106)(78 107)(79 108)(80 109)(96 134)(97 135)(98 131)(99 132)(100 133)(101 122)(102 123)(103 124)(104 125)(105 121)(111 130)(112 126)(113 127)(114 128)(115 129)(116 150)(117 146)(118 147)(119 148)(120 149)
(1 145 12 7)(2 141 13 8)(3 142 14 9)(4 143 15 10)(5 144 11 6)(16 55 137 41)(17 51 138 42)(18 52 139 43)(19 53 140 44)(20 54 136 45)(21 68 159 39)(22 69 160 40)(23 70 156 36)(24 66 157 37)(25 67 158 38)(26 35 50 151)(27 31 46 152)(28 32 47 153)(29 33 48 154)(30 34 49 155)(56 134 94 96)(57 135 95 97)(58 131 91 98)(59 132 92 99)(60 133 93 100)(61 122 82 101)(62 123 83 102)(63 124 84 103)(64 125 85 104)(65 121 81 105)(71 130 90 111)(72 126 86 112)(73 127 87 113)(74 128 88 114)(75 129 89 115)(76 150 110 116)(77 146 106 117)(78 147 107 118)(79 148 108 119)(80 149 109 120)
(1 106 66 81)(2 107 67 82)(3 108 68 83)(4 109 69 84)(5 110 70 85)(6 125 23 150)(7 121 24 146)(8 122 25 147)(9 123 21 148)(10 124 22 149)(11 76 36 64)(12 77 37 65)(13 78 38 61)(14 79 39 62)(15 80 40 63)(16 135 33 128)(17 131 34 129)(18 132 35 130)(19 133 31 126)(20 134 32 127)(26 59 43 71)(27 60 44 72)(28 56 45 73)(29 57 41 74)(30 58 42 75)(46 93 53 86)(47 94 54 87)(48 95 55 88)(49 91 51 89)(50 92 52 90)(96 153 113 136)(97 154 114 137)(98 155 115 138)(99 151 111 139)(100 152 112 140)(101 158 118 141)(102 159 119 142)(103 160 120 143)(104 156 116 144)(105 157 117 145)
G:=sub<Sym(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), (1,65)(2,61)(3,62)(4,63)(5,64)(6,133)(7,134)(8,135)(9,131)(10,132)(11,85)(12,81)(13,82)(14,83)(15,84)(16,122)(17,123)(18,124)(19,125)(20,121)(21,129)(22,130)(23,126)(24,127)(25,128)(26,90)(27,86)(28,87)(29,88)(30,89)(31,150)(32,146)(33,147)(34,148)(35,149)(36,110)(37,106)(38,107)(39,108)(40,109)(41,95)(42,91)(43,92)(44,93)(45,94)(46,72)(47,73)(48,74)(49,75)(50,71)(51,58)(52,59)(53,60)(54,56)(55,57)(66,77)(67,78)(68,79)(69,80)(70,76)(96,145)(97,141)(98,142)(99,143)(100,144)(101,137)(102,138)(103,139)(104,140)(105,136)(111,160)(112,156)(113,157)(114,158)(115,159)(116,152)(117,153)(118,154)(119,155)(120,151), (1,45)(2,41)(3,42)(4,43)(5,44)(6,140)(7,136)(8,137)(9,138)(10,139)(11,53)(12,54)(13,55)(14,51)(15,52)(16,141)(17,142)(18,143)(19,144)(20,145)(21,155)(22,151)(23,152)(24,153)(25,154)(26,69)(27,70)(28,66)(29,67)(30,68)(31,156)(32,157)(33,158)(34,159)(35,160)(36,46)(37,47)(38,48)(39,49)(40,50)(56,81)(57,82)(58,83)(59,84)(60,85)(61,95)(62,91)(63,92)(64,93)(65,94)(71,109)(72,110)(73,106)(74,107)(75,108)(76,86)(77,87)(78,88)(79,89)(80,90)(96,121)(97,122)(98,123)(99,124)(100,125)(101,135)(102,131)(103,132)(104,133)(105,134)(111,149)(112,150)(113,146)(114,147)(115,148)(116,126)(117,127)(118,128)(119,129)(120,130), (1,66)(2,67)(3,68)(4,69)(5,70)(6,23)(7,24)(8,25)(9,21)(10,22)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,34)(18,35)(19,31)(20,32)(26,43)(27,44)(28,45)(29,41)(30,42)(46,53)(47,54)(48,55)(49,51)(50,52)(56,73)(57,74)(58,75)(59,71)(60,72)(61,78)(62,79)(63,80)(64,76)(65,77)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,113)(97,114)(98,115)(99,111)(100,112)(101,118)(102,119)(103,120)(104,116)(105,117)(121,146)(122,147)(123,148)(124,149)(125,150)(126,133)(127,134)(128,135)(129,131)(130,132)(136,153)(137,154)(138,155)(139,151)(140,152)(141,158)(142,159)(143,160)(144,156)(145,157), (1,12)(2,13)(3,14)(4,15)(5,11)(6,144)(7,145)(8,141)(9,142)(10,143)(16,137)(17,138)(18,139)(19,140)(20,136)(21,159)(22,160)(23,156)(24,157)(25,158)(26,50)(27,46)(28,47)(29,48)(30,49)(31,152)(32,153)(33,154)(34,155)(35,151)(36,70)(37,66)(38,67)(39,68)(40,69)(41,55)(42,51)(43,52)(44,53)(45,54)(56,94)(57,95)(58,91)(59,92)(60,93)(61,82)(62,83)(63,84)(64,85)(65,81)(71,90)(72,86)(73,87)(74,88)(75,89)(76,110)(77,106)(78,107)(79,108)(80,109)(96,134)(97,135)(98,131)(99,132)(100,133)(101,122)(102,123)(103,124)(104,125)(105,121)(111,130)(112,126)(113,127)(114,128)(115,129)(116,150)(117,146)(118,147)(119,148)(120,149), (1,145,12,7)(2,141,13,8)(3,142,14,9)(4,143,15,10)(5,144,11,6)(16,55,137,41)(17,51,138,42)(18,52,139,43)(19,53,140,44)(20,54,136,45)(21,68,159,39)(22,69,160,40)(23,70,156,36)(24,66,157,37)(25,67,158,38)(26,35,50,151)(27,31,46,152)(28,32,47,153)(29,33,48,154)(30,34,49,155)(56,134,94,96)(57,135,95,97)(58,131,91,98)(59,132,92,99)(60,133,93,100)(61,122,82,101)(62,123,83,102)(63,124,84,103)(64,125,85,104)(65,121,81,105)(71,130,90,111)(72,126,86,112)(73,127,87,113)(74,128,88,114)(75,129,89,115)(76,150,110,116)(77,146,106,117)(78,147,107,118)(79,148,108,119)(80,149,109,120), (1,106,66,81)(2,107,67,82)(3,108,68,83)(4,109,69,84)(5,110,70,85)(6,125,23,150)(7,121,24,146)(8,122,25,147)(9,123,21,148)(10,124,22,149)(11,76,36,64)(12,77,37,65)(13,78,38,61)(14,79,39,62)(15,80,40,63)(16,135,33,128)(17,131,34,129)(18,132,35,130)(19,133,31,126)(20,134,32,127)(26,59,43,71)(27,60,44,72)(28,56,45,73)(29,57,41,74)(30,58,42,75)(46,93,53,86)(47,94,54,87)(48,95,55,88)(49,91,51,89)(50,92,52,90)(96,153,113,136)(97,154,114,137)(98,155,115,138)(99,151,111,139)(100,152,112,140)(101,158,118,141)(102,159,119,142)(103,160,120,143)(104,156,116,144)(105,157,117,145)>;
G:=Group( (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), (1,65)(2,61)(3,62)(4,63)(5,64)(6,133)(7,134)(8,135)(9,131)(10,132)(11,85)(12,81)(13,82)(14,83)(15,84)(16,122)(17,123)(18,124)(19,125)(20,121)(21,129)(22,130)(23,126)(24,127)(25,128)(26,90)(27,86)(28,87)(29,88)(30,89)(31,150)(32,146)(33,147)(34,148)(35,149)(36,110)(37,106)(38,107)(39,108)(40,109)(41,95)(42,91)(43,92)(44,93)(45,94)(46,72)(47,73)(48,74)(49,75)(50,71)(51,58)(52,59)(53,60)(54,56)(55,57)(66,77)(67,78)(68,79)(69,80)(70,76)(96,145)(97,141)(98,142)(99,143)(100,144)(101,137)(102,138)(103,139)(104,140)(105,136)(111,160)(112,156)(113,157)(114,158)(115,159)(116,152)(117,153)(118,154)(119,155)(120,151), (1,45)(2,41)(3,42)(4,43)(5,44)(6,140)(7,136)(8,137)(9,138)(10,139)(11,53)(12,54)(13,55)(14,51)(15,52)(16,141)(17,142)(18,143)(19,144)(20,145)(21,155)(22,151)(23,152)(24,153)(25,154)(26,69)(27,70)(28,66)(29,67)(30,68)(31,156)(32,157)(33,158)(34,159)(35,160)(36,46)(37,47)(38,48)(39,49)(40,50)(56,81)(57,82)(58,83)(59,84)(60,85)(61,95)(62,91)(63,92)(64,93)(65,94)(71,109)(72,110)(73,106)(74,107)(75,108)(76,86)(77,87)(78,88)(79,89)(80,90)(96,121)(97,122)(98,123)(99,124)(100,125)(101,135)(102,131)(103,132)(104,133)(105,134)(111,149)(112,150)(113,146)(114,147)(115,148)(116,126)(117,127)(118,128)(119,129)(120,130), (1,66)(2,67)(3,68)(4,69)(5,70)(6,23)(7,24)(8,25)(9,21)(10,22)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,34)(18,35)(19,31)(20,32)(26,43)(27,44)(28,45)(29,41)(30,42)(46,53)(47,54)(48,55)(49,51)(50,52)(56,73)(57,74)(58,75)(59,71)(60,72)(61,78)(62,79)(63,80)(64,76)(65,77)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,113)(97,114)(98,115)(99,111)(100,112)(101,118)(102,119)(103,120)(104,116)(105,117)(121,146)(122,147)(123,148)(124,149)(125,150)(126,133)(127,134)(128,135)(129,131)(130,132)(136,153)(137,154)(138,155)(139,151)(140,152)(141,158)(142,159)(143,160)(144,156)(145,157), (1,12)(2,13)(3,14)(4,15)(5,11)(6,144)(7,145)(8,141)(9,142)(10,143)(16,137)(17,138)(18,139)(19,140)(20,136)(21,159)(22,160)(23,156)(24,157)(25,158)(26,50)(27,46)(28,47)(29,48)(30,49)(31,152)(32,153)(33,154)(34,155)(35,151)(36,70)(37,66)(38,67)(39,68)(40,69)(41,55)(42,51)(43,52)(44,53)(45,54)(56,94)(57,95)(58,91)(59,92)(60,93)(61,82)(62,83)(63,84)(64,85)(65,81)(71,90)(72,86)(73,87)(74,88)(75,89)(76,110)(77,106)(78,107)(79,108)(80,109)(96,134)(97,135)(98,131)(99,132)(100,133)(101,122)(102,123)(103,124)(104,125)(105,121)(111,130)(112,126)(113,127)(114,128)(115,129)(116,150)(117,146)(118,147)(119,148)(120,149), (1,145,12,7)(2,141,13,8)(3,142,14,9)(4,143,15,10)(5,144,11,6)(16,55,137,41)(17,51,138,42)(18,52,139,43)(19,53,140,44)(20,54,136,45)(21,68,159,39)(22,69,160,40)(23,70,156,36)(24,66,157,37)(25,67,158,38)(26,35,50,151)(27,31,46,152)(28,32,47,153)(29,33,48,154)(30,34,49,155)(56,134,94,96)(57,135,95,97)(58,131,91,98)(59,132,92,99)(60,133,93,100)(61,122,82,101)(62,123,83,102)(63,124,84,103)(64,125,85,104)(65,121,81,105)(71,130,90,111)(72,126,86,112)(73,127,87,113)(74,128,88,114)(75,129,89,115)(76,150,110,116)(77,146,106,117)(78,147,107,118)(79,148,108,119)(80,149,109,120), (1,106,66,81)(2,107,67,82)(3,108,68,83)(4,109,69,84)(5,110,70,85)(6,125,23,150)(7,121,24,146)(8,122,25,147)(9,123,21,148)(10,124,22,149)(11,76,36,64)(12,77,37,65)(13,78,38,61)(14,79,39,62)(15,80,40,63)(16,135,33,128)(17,131,34,129)(18,132,35,130)(19,133,31,126)(20,134,32,127)(26,59,43,71)(27,60,44,72)(28,56,45,73)(29,57,41,74)(30,58,42,75)(46,93,53,86)(47,94,54,87)(48,95,55,88)(49,91,51,89)(50,92,52,90)(96,153,113,136)(97,154,114,137)(98,155,115,138)(99,151,111,139)(100,152,112,140)(101,158,118,141)(102,159,119,142)(103,160,120,143)(104,156,116,144)(105,157,117,145) );
G=PermutationGroup([[(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)], [(1,65),(2,61),(3,62),(4,63),(5,64),(6,133),(7,134),(8,135),(9,131),(10,132),(11,85),(12,81),(13,82),(14,83),(15,84),(16,122),(17,123),(18,124),(19,125),(20,121),(21,129),(22,130),(23,126),(24,127),(25,128),(26,90),(27,86),(28,87),(29,88),(30,89),(31,150),(32,146),(33,147),(34,148),(35,149),(36,110),(37,106),(38,107),(39,108),(40,109),(41,95),(42,91),(43,92),(44,93),(45,94),(46,72),(47,73),(48,74),(49,75),(50,71),(51,58),(52,59),(53,60),(54,56),(55,57),(66,77),(67,78),(68,79),(69,80),(70,76),(96,145),(97,141),(98,142),(99,143),(100,144),(101,137),(102,138),(103,139),(104,140),(105,136),(111,160),(112,156),(113,157),(114,158),(115,159),(116,152),(117,153),(118,154),(119,155),(120,151)], [(1,45),(2,41),(3,42),(4,43),(5,44),(6,140),(7,136),(8,137),(9,138),(10,139),(11,53),(12,54),(13,55),(14,51),(15,52),(16,141),(17,142),(18,143),(19,144),(20,145),(21,155),(22,151),(23,152),(24,153),(25,154),(26,69),(27,70),(28,66),(29,67),(30,68),(31,156),(32,157),(33,158),(34,159),(35,160),(36,46),(37,47),(38,48),(39,49),(40,50),(56,81),(57,82),(58,83),(59,84),(60,85),(61,95),(62,91),(63,92),(64,93),(65,94),(71,109),(72,110),(73,106),(74,107),(75,108),(76,86),(77,87),(78,88),(79,89),(80,90),(96,121),(97,122),(98,123),(99,124),(100,125),(101,135),(102,131),(103,132),(104,133),(105,134),(111,149),(112,150),(113,146),(114,147),(115,148),(116,126),(117,127),(118,128),(119,129),(120,130)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,23),(7,24),(8,25),(9,21),(10,22),(11,36),(12,37),(13,38),(14,39),(15,40),(16,33),(17,34),(18,35),(19,31),(20,32),(26,43),(27,44),(28,45),(29,41),(30,42),(46,53),(47,54),(48,55),(49,51),(50,52),(56,73),(57,74),(58,75),(59,71),(60,72),(61,78),(62,79),(63,80),(64,76),(65,77),(81,106),(82,107),(83,108),(84,109),(85,110),(86,93),(87,94),(88,95),(89,91),(90,92),(96,113),(97,114),(98,115),(99,111),(100,112),(101,118),(102,119),(103,120),(104,116),(105,117),(121,146),(122,147),(123,148),(124,149),(125,150),(126,133),(127,134),(128,135),(129,131),(130,132),(136,153),(137,154),(138,155),(139,151),(140,152),(141,158),(142,159),(143,160),(144,156),(145,157)], [(1,12),(2,13),(3,14),(4,15),(5,11),(6,144),(7,145),(8,141),(9,142),(10,143),(16,137),(17,138),(18,139),(19,140),(20,136),(21,159),(22,160),(23,156),(24,157),(25,158),(26,50),(27,46),(28,47),(29,48),(30,49),(31,152),(32,153),(33,154),(34,155),(35,151),(36,70),(37,66),(38,67),(39,68),(40,69),(41,55),(42,51),(43,52),(44,53),(45,54),(56,94),(57,95),(58,91),(59,92),(60,93),(61,82),(62,83),(63,84),(64,85),(65,81),(71,90),(72,86),(73,87),(74,88),(75,89),(76,110),(77,106),(78,107),(79,108),(80,109),(96,134),(97,135),(98,131),(99,132),(100,133),(101,122),(102,123),(103,124),(104,125),(105,121),(111,130),(112,126),(113,127),(114,128),(115,129),(116,150),(117,146),(118,147),(119,148),(120,149)], [(1,145,12,7),(2,141,13,8),(3,142,14,9),(4,143,15,10),(5,144,11,6),(16,55,137,41),(17,51,138,42),(18,52,139,43),(19,53,140,44),(20,54,136,45),(21,68,159,39),(22,69,160,40),(23,70,156,36),(24,66,157,37),(25,67,158,38),(26,35,50,151),(27,31,46,152),(28,32,47,153),(29,33,48,154),(30,34,49,155),(56,134,94,96),(57,135,95,97),(58,131,91,98),(59,132,92,99),(60,133,93,100),(61,122,82,101),(62,123,83,102),(63,124,84,103),(64,125,85,104),(65,121,81,105),(71,130,90,111),(72,126,86,112),(73,127,87,113),(74,128,88,114),(75,129,89,115),(76,150,110,116),(77,146,106,117),(78,147,107,118),(79,148,108,119),(80,149,109,120)], [(1,106,66,81),(2,107,67,82),(3,108,68,83),(4,109,69,84),(5,110,70,85),(6,125,23,150),(7,121,24,146),(8,122,25,147),(9,123,21,148),(10,124,22,149),(11,76,36,64),(12,77,37,65),(13,78,38,61),(14,79,39,62),(15,80,40,63),(16,135,33,128),(17,131,34,129),(18,132,35,130),(19,133,31,126),(20,134,32,127),(26,59,43,71),(27,60,44,72),(28,56,45,73),(29,57,41,74),(30,58,42,75),(46,93,53,86),(47,94,54,87),(48,95,55,88),(49,91,51,89),(50,92,52,90),(96,153,113,136),(97,154,114,137),(98,155,115,138),(99,151,111,139),(100,152,112,140),(101,158,118,141),(102,159,119,142),(103,160,120,143),(104,156,116,144),(105,157,117,145)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AV | 20AW | ··· | 20BL |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | D4 | C4○D4 | C5×D4 | C5×C4○D4 |
| kernel | C5×C24.3C22 | C2×C4×C20 | C10×C22⋊C4 | C10×C4⋊C4 | D4×C2×C10 | D4×C10 | C24.3C22 | C2×C42 | C2×C22⋊C4 | C2×C4⋊C4 | C22×D4 | C2×D4 | C2×C20 | C2×C10 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 8 | 4 | 4 | 16 | 4 | 4 | 32 | 8 | 4 | 32 | 16 |
Matrix representation of C5×C24.3C22 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 39 |
| 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 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 32 | 18 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 39 | 0 | 0 |
| 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 1 | 40 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,40,0,0,0,0,2,1,0,0,0,0,0,1,0,0,0,0,39,40],[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],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,32,0,0,0,0,18,9,0,0,0,0,0,40,40,0,0,0,0,1],[1,0,0,0,0,0,1,1,0,0,0,39,40,0,0,0,0,0,1,1,0,0,0,39,40] >;
C5×C24.3C22 in GAP, Magma, Sage, TeX
C_5\times C_2^4._3C_2^2
% in TeX
G:=Group("C5xC2^4.3C2^2"); // GroupNames label
G:=SmallGroup(320,891);
// by ID
G=gap.SmallGroup(320,891);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1408,1766,436]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=e^2=1,f^2=e,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,f*b*f^-1=b*c=c*b,g*b*g^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e>;
// generators/relations