direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C22.26C24, (C4×D4)⋊9C10, (C2×C20)⋊34D4, C4⋊Q8⋊19C10, (D4×C20)⋊38C2, C4.15(D4×C10), C20⋊13(C4○D4), C4⋊D4⋊20C10, C4⋊1D4⋊10C10, (C2×C42)⋊12C10, C20.322(C2×D4), C22.1(D4×C10), C4.4D4⋊18C10, C42.90(C2×C10), (C2×C10).352C24, (C2×C20).661C23, (C4×C20).375C22, C10.188(C22×D4), (D4×C10).319C22, C23.34(C22×C10), C22.26(C23×C10), (C22×C10).89C23, (Q8×C10).268C22, (C22×C20).597C22, (C2×C4×C20)⋊25C2, C4⋊1(C5×C4○D4), (C2×C4)⋊8(C5×D4), (C5×C4⋊Q8)⋊40C2, C2.12(D4×C2×C10), (C2×C4○D4)⋊3C10, (C10×C4○D4)⋊19C2, (C5×C4⋊D4)⋊47C2, C4⋊C4.66(C2×C10), (C5×C4⋊1D4)⋊21C2, C2.13(C10×C4○D4), (C2×C10).89(C2×D4), (C2×D4).64(C2×C10), C10.232(C2×C4○D4), (C5×C4.4D4)⋊38C2, (C2×Q8).55(C2×C10), C22⋊C4.13(C2×C10), (C5×C4⋊C4).389C22, (C2×C4).19(C22×C10), (C22×C4).56(C2×C10), (C5×C22⋊C4).147C22, SmallGroup(320,1534)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.26C24
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=e2=1, f2=c, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >
Subgroups: 466 in 310 conjugacy classes, 170 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C22.26C24, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C2×C4×C20, D4×C20, C5×C4⋊D4, C5×C4.4D4, C5×C4⋊1D4, C5×C4⋊Q8, C10×C4○D4, C5×C22.26C24
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C24, C2×C10, C22×D4, C2×C4○D4, C5×D4, C22×C10, C22.26C24, D4×C10, C5×C4○D4, C23×C10, D4×C2×C10, C10×C4○D4, C5×C22.26C24
►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 35)(2 31)(3 32)(4 33)(5 34)(6 16)(7 17)(8 18)(9 19)(10 20)(11 156)(12 157)(13 158)(14 159)(15 160)(21 28)(22 29)(23 30)(24 26)(25 27)(36 55)(37 51)(38 52)(39 53)(40 54)(41 48)(42 49)(43 50)(44 46)(45 47)(56 75)(57 71)(58 72)(59 73)(60 74)(61 68)(62 69)(63 70)(64 66)(65 67)(76 95)(77 91)(78 92)(79 93)(80 94)(81 88)(82 89)(83 90)(84 86)(85 87)(96 115)(97 111)(98 112)(99 113)(100 114)(101 108)(102 109)(103 110)(104 106)(105 107)(116 135)(117 131)(118 132)(119 133)(120 134)(121 128)(122 129)(123 130)(124 126)(125 127)(136 155)(137 151)(138 152)(139 153)(140 154)(141 148)(142 149)(143 150)(144 146)(145 147)
(1 30)(2 26)(3 27)(4 28)(5 29)(6 156)(7 157)(8 158)(9 159)(10 160)(11 16)(12 17)(13 18)(14 19)(15 20)(21 33)(22 34)(23 35)(24 31)(25 32)(36 50)(37 46)(38 47)(39 48)(40 49)(41 53)(42 54)(43 55)(44 51)(45 52)(56 70)(57 66)(58 67)(59 68)(60 69)(61 73)(62 74)(63 75)(64 71)(65 72)(76 90)(77 86)(78 87)(79 88)(80 89)(81 93)(82 94)(83 95)(84 91)(85 92)(96 110)(97 106)(98 107)(99 108)(100 109)(101 113)(102 114)(103 115)(104 111)(105 112)(116 130)(117 126)(118 127)(119 128)(120 129)(121 133)(122 134)(123 135)(124 131)(125 132)(136 150)(137 146)(138 147)(139 148)(140 149)(141 153)(142 154)(143 155)(144 151)(145 152)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 123)(7 124)(8 125)(9 121)(10 122)(11 116)(12 117)(13 118)(14 119)(15 120)(16 130)(17 126)(18 127)(19 128)(20 129)(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 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(96 155)(97 151)(98 152)(99 153)(100 154)(101 148)(102 149)(103 150)(104 146)(105 147)(106 144)(107 145)(108 141)(109 142)(110 143)(111 137)(112 138)(113 139)(114 140)(115 136)(131 157)(132 158)(133 159)(134 160)(135 156)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 90)(7 86)(8 87)(9 88)(10 89)(11 95)(12 91)(13 92)(14 93)(15 94)(16 83)(17 84)(18 85)(19 81)(20 82)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
(1 70 30 56)(2 66 26 57)(3 67 27 58)(4 68 28 59)(5 69 29 60)(6 130 156 116)(7 126 157 117)(8 127 158 118)(9 128 159 119)(10 129 160 120)(11 135 16 123)(12 131 17 124)(13 132 18 125)(14 133 19 121)(15 134 20 122)(21 73 33 61)(22 74 34 62)(23 75 35 63)(24 71 31 64)(25 72 32 65)(36 90 50 76)(37 86 46 77)(38 87 47 78)(39 88 48 79)(40 89 49 80)(41 93 53 81)(42 94 54 82)(43 95 55 83)(44 91 51 84)(45 92 52 85)(96 150 110 136)(97 146 106 137)(98 147 107 138)(99 148 108 139)(100 149 109 140)(101 153 113 141)(102 154 114 142)(103 155 115 143)(104 151 111 144)(105 152 112 145)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 150 16 143)(7 146 17 144)(8 147 18 145)(9 148 19 141)(10 149 20 142)(11 155 156 136)(12 151 157 137)(13 152 158 138)(14 153 159 139)(15 154 160 140)(21 48 28 41)(22 49 29 42)(23 50 30 43)(24 46 26 44)(25 47 27 45)(56 95 75 76)(57 91 71 77)(58 92 72 78)(59 93 73 79)(60 94 74 80)(61 88 68 81)(62 89 69 82)(63 90 70 83)(64 86 66 84)(65 87 67 85)(96 135 115 116)(97 131 111 117)(98 132 112 118)(99 133 113 119)(100 134 114 120)(101 128 108 121)(102 129 109 122)(103 130 110 123)(104 126 106 124)(105 127 107 125)
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), (1,35)(2,31)(3,32)(4,33)(5,34)(6,16)(7,17)(8,18)(9,19)(10,20)(11,156)(12,157)(13,158)(14,159)(15,160)(21,28)(22,29)(23,30)(24,26)(25,27)(36,55)(37,51)(38,52)(39,53)(40,54)(41,48)(42,49)(43,50)(44,46)(45,47)(56,75)(57,71)(58,72)(59,73)(60,74)(61,68)(62,69)(63,70)(64,66)(65,67)(76,95)(77,91)(78,92)(79,93)(80,94)(81,88)(82,89)(83,90)(84,86)(85,87)(96,115)(97,111)(98,112)(99,113)(100,114)(101,108)(102,109)(103,110)(104,106)(105,107)(116,135)(117,131)(118,132)(119,133)(120,134)(121,128)(122,129)(123,130)(124,126)(125,127)(136,155)(137,151)(138,152)(139,153)(140,154)(141,148)(142,149)(143,150)(144,146)(145,147), (1,30)(2,26)(3,27)(4,28)(5,29)(6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(21,33)(22,34)(23,35)(24,31)(25,32)(36,50)(37,46)(38,47)(39,48)(40,49)(41,53)(42,54)(43,55)(44,51)(45,52)(56,70)(57,66)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,71)(65,72)(76,90)(77,86)(78,87)(79,88)(80,89)(81,93)(82,94)(83,95)(84,91)(85,92)(96,110)(97,106)(98,107)(99,108)(100,109)(101,113)(102,114)(103,115)(104,111)(105,112)(116,130)(117,126)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,131)(125,132)(136,150)(137,146)(138,147)(139,148)(140,149)(141,153)(142,154)(143,155)(144,151)(145,152), (1,56)(2,57)(3,58)(4,59)(5,60)(6,123)(7,124)(8,125)(9,121)(10,122)(11,116)(12,117)(13,118)(14,119)(15,120)(16,130)(17,126)(18,127)(19,128)(20,129)(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)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(96,155)(97,151)(98,152)(99,153)(100,154)(101,148)(102,149)(103,150)(104,146)(105,147)(106,144)(107,145)(108,141)(109,142)(110,143)(111,137)(112,138)(113,139)(114,140)(115,136)(131,157)(132,158)(133,159)(134,160)(135,156), (1,96)(2,97)(3,98)(4,99)(5,100)(6,90)(7,86)(8,87)(9,88)(10,89)(11,95)(12,91)(13,92)(14,93)(15,94)(16,83)(17,84)(18,85)(19,81)(20,82)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,70,30,56)(2,66,26,57)(3,67,27,58)(4,68,28,59)(5,69,29,60)(6,130,156,116)(7,126,157,117)(8,127,158,118)(9,128,159,119)(10,129,160,120)(11,135,16,123)(12,131,17,124)(13,132,18,125)(14,133,19,121)(15,134,20,122)(21,73,33,61)(22,74,34,62)(23,75,35,63)(24,71,31,64)(25,72,32,65)(36,90,50,76)(37,86,46,77)(38,87,47,78)(39,88,48,79)(40,89,49,80)(41,93,53,81)(42,94,54,82)(43,95,55,83)(44,91,51,84)(45,92,52,85)(96,150,110,136)(97,146,106,137)(98,147,107,138)(99,148,108,139)(100,149,109,140)(101,153,113,141)(102,154,114,142)(103,155,115,143)(104,151,111,144)(105,152,112,145), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,150,16,143)(7,146,17,144)(8,147,18,145)(9,148,19,141)(10,149,20,142)(11,155,156,136)(12,151,157,137)(13,152,158,138)(14,153,159,139)(15,154,160,140)(21,48,28,41)(22,49,29,42)(23,50,30,43)(24,46,26,44)(25,47,27,45)(56,95,75,76)(57,91,71,77)(58,92,72,78)(59,93,73,79)(60,94,74,80)(61,88,68,81)(62,89,69,82)(63,90,70,83)(64,86,66,84)(65,87,67,85)(96,135,115,116)(97,131,111,117)(98,132,112,118)(99,133,113,119)(100,134,114,120)(101,128,108,121)(102,129,109,122)(103,130,110,123)(104,126,106,124)(105,127,107,125)>;
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), (1,35)(2,31)(3,32)(4,33)(5,34)(6,16)(7,17)(8,18)(9,19)(10,20)(11,156)(12,157)(13,158)(14,159)(15,160)(21,28)(22,29)(23,30)(24,26)(25,27)(36,55)(37,51)(38,52)(39,53)(40,54)(41,48)(42,49)(43,50)(44,46)(45,47)(56,75)(57,71)(58,72)(59,73)(60,74)(61,68)(62,69)(63,70)(64,66)(65,67)(76,95)(77,91)(78,92)(79,93)(80,94)(81,88)(82,89)(83,90)(84,86)(85,87)(96,115)(97,111)(98,112)(99,113)(100,114)(101,108)(102,109)(103,110)(104,106)(105,107)(116,135)(117,131)(118,132)(119,133)(120,134)(121,128)(122,129)(123,130)(124,126)(125,127)(136,155)(137,151)(138,152)(139,153)(140,154)(141,148)(142,149)(143,150)(144,146)(145,147), (1,30)(2,26)(3,27)(4,28)(5,29)(6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(21,33)(22,34)(23,35)(24,31)(25,32)(36,50)(37,46)(38,47)(39,48)(40,49)(41,53)(42,54)(43,55)(44,51)(45,52)(56,70)(57,66)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,71)(65,72)(76,90)(77,86)(78,87)(79,88)(80,89)(81,93)(82,94)(83,95)(84,91)(85,92)(96,110)(97,106)(98,107)(99,108)(100,109)(101,113)(102,114)(103,115)(104,111)(105,112)(116,130)(117,126)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,131)(125,132)(136,150)(137,146)(138,147)(139,148)(140,149)(141,153)(142,154)(143,155)(144,151)(145,152), (1,56)(2,57)(3,58)(4,59)(5,60)(6,123)(7,124)(8,125)(9,121)(10,122)(11,116)(12,117)(13,118)(14,119)(15,120)(16,130)(17,126)(18,127)(19,128)(20,129)(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)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(96,155)(97,151)(98,152)(99,153)(100,154)(101,148)(102,149)(103,150)(104,146)(105,147)(106,144)(107,145)(108,141)(109,142)(110,143)(111,137)(112,138)(113,139)(114,140)(115,136)(131,157)(132,158)(133,159)(134,160)(135,156), (1,96)(2,97)(3,98)(4,99)(5,100)(6,90)(7,86)(8,87)(9,88)(10,89)(11,95)(12,91)(13,92)(14,93)(15,94)(16,83)(17,84)(18,85)(19,81)(20,82)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,70,30,56)(2,66,26,57)(3,67,27,58)(4,68,28,59)(5,69,29,60)(6,130,156,116)(7,126,157,117)(8,127,158,118)(9,128,159,119)(10,129,160,120)(11,135,16,123)(12,131,17,124)(13,132,18,125)(14,133,19,121)(15,134,20,122)(21,73,33,61)(22,74,34,62)(23,75,35,63)(24,71,31,64)(25,72,32,65)(36,90,50,76)(37,86,46,77)(38,87,47,78)(39,88,48,79)(40,89,49,80)(41,93,53,81)(42,94,54,82)(43,95,55,83)(44,91,51,84)(45,92,52,85)(96,150,110,136)(97,146,106,137)(98,147,107,138)(99,148,108,139)(100,149,109,140)(101,153,113,141)(102,154,114,142)(103,155,115,143)(104,151,111,144)(105,152,112,145), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,150,16,143)(7,146,17,144)(8,147,18,145)(9,148,19,141)(10,149,20,142)(11,155,156,136)(12,151,157,137)(13,152,158,138)(14,153,159,139)(15,154,160,140)(21,48,28,41)(22,49,29,42)(23,50,30,43)(24,46,26,44)(25,47,27,45)(56,95,75,76)(57,91,71,77)(58,92,72,78)(59,93,73,79)(60,94,74,80)(61,88,68,81)(62,89,69,82)(63,90,70,83)(64,86,66,84)(65,87,67,85)(96,135,115,116)(97,131,111,117)(98,132,112,118)(99,133,113,119)(100,134,114,120)(101,128,108,121)(102,129,109,122)(103,130,110,123)(104,126,106,124)(105,127,107,125) );
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)], [(1,35),(2,31),(3,32),(4,33),(5,34),(6,16),(7,17),(8,18),(9,19),(10,20),(11,156),(12,157),(13,158),(14,159),(15,160),(21,28),(22,29),(23,30),(24,26),(25,27),(36,55),(37,51),(38,52),(39,53),(40,54),(41,48),(42,49),(43,50),(44,46),(45,47),(56,75),(57,71),(58,72),(59,73),(60,74),(61,68),(62,69),(63,70),(64,66),(65,67),(76,95),(77,91),(78,92),(79,93),(80,94),(81,88),(82,89),(83,90),(84,86),(85,87),(96,115),(97,111),(98,112),(99,113),(100,114),(101,108),(102,109),(103,110),(104,106),(105,107),(116,135),(117,131),(118,132),(119,133),(120,134),(121,128),(122,129),(123,130),(124,126),(125,127),(136,155),(137,151),(138,152),(139,153),(140,154),(141,148),(142,149),(143,150),(144,146),(145,147)], [(1,30),(2,26),(3,27),(4,28),(5,29),(6,156),(7,157),(8,158),(9,159),(10,160),(11,16),(12,17),(13,18),(14,19),(15,20),(21,33),(22,34),(23,35),(24,31),(25,32),(36,50),(37,46),(38,47),(39,48),(40,49),(41,53),(42,54),(43,55),(44,51),(45,52),(56,70),(57,66),(58,67),(59,68),(60,69),(61,73),(62,74),(63,75),(64,71),(65,72),(76,90),(77,86),(78,87),(79,88),(80,89),(81,93),(82,94),(83,95),(84,91),(85,92),(96,110),(97,106),(98,107),(99,108),(100,109),(101,113),(102,114),(103,115),(104,111),(105,112),(116,130),(117,126),(118,127),(119,128),(120,129),(121,133),(122,134),(123,135),(124,131),(125,132),(136,150),(137,146),(138,147),(139,148),(140,149),(141,153),(142,154),(143,155),(144,151),(145,152)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,123),(7,124),(8,125),(9,121),(10,122),(11,116),(12,117),(13,118),(14,119),(15,120),(16,130),(17,126),(18,127),(19,128),(20,129),(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),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(96,155),(97,151),(98,152),(99,153),(100,154),(101,148),(102,149),(103,150),(104,146),(105,147),(106,144),(107,145),(108,141),(109,142),(110,143),(111,137),(112,138),(113,139),(114,140),(115,136),(131,157),(132,158),(133,159),(134,160),(135,156)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,90),(7,86),(8,87),(9,88),(10,89),(11,95),(12,91),(13,92),(14,93),(15,94),(16,83),(17,84),(18,85),(19,81),(20,82),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)], [(1,70,30,56),(2,66,26,57),(3,67,27,58),(4,68,28,59),(5,69,29,60),(6,130,156,116),(7,126,157,117),(8,127,158,118),(9,128,159,119),(10,129,160,120),(11,135,16,123),(12,131,17,124),(13,132,18,125),(14,133,19,121),(15,134,20,122),(21,73,33,61),(22,74,34,62),(23,75,35,63),(24,71,31,64),(25,72,32,65),(36,90,50,76),(37,86,46,77),(38,87,47,78),(39,88,48,79),(40,89,49,80),(41,93,53,81),(42,94,54,82),(43,95,55,83),(44,91,51,84),(45,92,52,85),(96,150,110,136),(97,146,106,137),(98,147,107,138),(99,148,108,139),(100,149,109,140),(101,153,113,141),(102,154,114,142),(103,155,115,143),(104,151,111,144),(105,152,112,145)], [(1,55,35,36),(2,51,31,37),(3,52,32,38),(4,53,33,39),(5,54,34,40),(6,150,16,143),(7,146,17,144),(8,147,18,145),(9,148,19,141),(10,149,20,142),(11,155,156,136),(12,151,157,137),(13,152,158,138),(14,153,159,139),(15,154,160,140),(21,48,28,41),(22,49,29,42),(23,50,30,43),(24,46,26,44),(25,47,27,45),(56,95,75,76),(57,91,71,77),(58,92,72,78),(59,93,73,79),(60,94,74,80),(61,88,68,81),(62,89,69,82),(63,90,70,83),(64,86,66,84),(65,87,67,85),(96,135,115,116),(97,131,111,117),(98,132,112,118),(99,133,113,119),(100,134,114,120),(101,128,108,121),(102,129,109,122),(103,130,110,123),(104,126,106,124),(105,127,107,125)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AJ | 20A | ··· | 20P | 20Q | ··· | 20BD | 20BE | ··· | 20BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | C4○D4 | C5×D4 | C5×C4○D4 |
| kernel | C5×C22.26C24 | C2×C4×C20 | D4×C20 | C5×C4⋊D4 | C5×C4.4D4 | C5×C4⋊1D4 | C5×C4⋊Q8 | C10×C4○D4 | C22.26C24 | C2×C42 | C4×D4 | C4⋊D4 | C4.4D4 | C4⋊1D4 | C4⋊Q8 | C2×C4○D4 | C2×C20 | C20 | C2×C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 4 | 4 | 16 | 16 | 8 | 4 | 4 | 8 | 4 | 8 | 16 | 32 |
Matrix representation of C5×C22.26C24 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 9 | 0 | 0 |
| 32 | 0 | 0 | 0 |
| 0 | 0 | 24 | 39 |
| 0 | 0 | 21 | 17 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,40,17,0,0,0,1],[0,32,0,0,9,0,0,0,0,0,24,21,0,0,39,17],[0,40,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,0,32,0,0,0,0,9,0,0,0,0,9] >;
C5×C22.26C24 in GAP, Magma, Sage, TeX
C_5\times C_2^2._{26}C_2^4 % in TeX
G:=Group("C5xC2^2.26C2^4"); // GroupNames label
G:=SmallGroup(320,1534);
// by ID
G=gap.SmallGroup(320,1534);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,856,304]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=e^2=1,f^2=c,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations