direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C22.47C24, C10.1642+ 1+4, (D4×C20)⋊49C2, (C4×D4)⋊20C10, C4⋊D4⋊15C10, C42.C2⋊9C10, C42⋊2C2⋊6C10, C42.47(C2×C10), C42⋊C2⋊16C10, C20.324(C4○D4), (C4×C20).288C22, (C2×C10).373C24, (C2×C20).963C23, (D4×C10).221C22, C22.D4⋊11C10, C22.47(C23×C10), C23.19(C22×C10), C2.16(C5×2+ 1+4), (C22×C20).458C22, (C22×C10).102C23, (C10×C4⋊C4)⋊49C2, (C2×C4⋊C4)⋊22C10, C4.36(C5×C4○D4), (C5×C4⋊D4)⋊42C2, C4⋊C4.73(C2×C10), C2.26(C10×C4○D4), (C2×D4).34(C2×C10), C10.245(C2×C4○D4), (C5×C42.C2)⋊26C2, C22.11(C5×C4○D4), (C5×C42⋊C2)⋊37C2, (C5×C42⋊2C2)⋊17C2, C22⋊C4.23(C2×C10), (C5×C4⋊C4).399C22, (C22×C4).12(C2×C10), (C2×C4).62(C22×C10), (C2×C10).179(C4○D4), (C5×C22.D4)⋊30C2, (C5×C22⋊C4).155C22, SmallGroup(320,1555)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.47C24
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=f2=1, e2=cb=bc, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
Subgroups: 362 in 238 conjugacy classes, 150 normal (62 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×11], C5, C2×C4 [×5], C2×C4 [×6], C2×C4 [×8], D4 [×10], C23 [×2], C23 [×2], C10 [×3], C10 [×5], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C20 [×2], C20 [×10], C2×C10, C2×C10 [×2], C2×C10 [×11], C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C2×C20 [×5], C2×C20 [×6], C2×C20 [×8], C5×D4 [×10], C22×C10 [×2], C22×C10 [×2], C22.47C24, C4×C20, C4×C20 [×2], C5×C22⋊C4 [×2], C5×C22⋊C4 [×8], C5×C4⋊C4 [×4], C5×C4⋊C4 [×6], C22×C20 [×2], C22×C20 [×4], D4×C10 [×2], D4×C10 [×4], C10×C4⋊C4, C5×C42⋊C2, D4×C20 [×2], D4×C20 [×2], C5×C4⋊D4 [×2], C5×C4⋊D4 [×2], C5×C22.D4 [×2], C5×C42.C2, C5×C42⋊2C2 [×2], C5×C22.47C24
Quotients: C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C4○D4 [×4], C24, C2×C10 [×35], C2×C4○D4 [×2], 2+ 1+4, C22×C10 [×15], C22.47C24, C5×C4○D4 [×4], C23×C10, C10×C4○D4 [×2], C5×2+ 1+4, C5×C22.47C24
►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)(6 160)(7 156)(8 157)(9 158)(10 159)(11 17)(12 18)(13 19)(14 20)(15 16)(21 27)(22 28)(23 29)(24 30)(25 26)(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)(76 95)(77 91)(78 92)(79 93)(80 94)(81 87)(82 88)(83 89)(84 90)(85 86)(96 115)(97 111)(98 112)(99 113)(100 114)(101 107)(102 108)(103 109)(104 110)(105 106)(116 135)(117 131)(118 132)(119 133)(120 134)(121 127)(122 128)(123 129)(124 130)(125 126)(136 155)(137 151)(138 152)(139 153)(140 154)(141 147)(142 148)(143 149)(144 150)(145 146)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 157)(12 158)(13 159)(14 160)(15 156)(26 34)(27 35)(28 31)(29 32)(30 33)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)(56 61)(57 62)(58 63)(59 64)(60 65)(66 74)(67 75)(68 71)(69 72)(70 73)(76 81)(77 82)(78 83)(79 84)(80 85)(86 94)(87 95)(88 91)(89 92)(90 93)(96 101)(97 102)(98 103)(99 104)(100 105)(106 114)(107 115)(108 111)(109 112)(110 113)(116 121)(117 122)(118 123)(119 124)(120 125)(126 134)(127 135)(128 131)(129 132)(130 133)(136 141)(137 142)(138 143)(139 144)(140 145)(146 154)(147 155)(148 151)(149 152)(150 153)
(1 136)(2 137)(3 138)(4 139)(5 140)(6 40)(7 36)(8 37)(9 38)(10 39)(11 48)(12 49)(13 50)(14 46)(15 47)(16 41)(17 42)(18 43)(19 44)(20 45)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(51 157)(52 158)(53 159)(54 160)(55 156)(56 101)(57 102)(58 103)(59 104)(60 105)(61 96)(62 97)(63 98)(64 99)(65 100)(66 114)(67 115)(68 111)(69 112)(70 113)(71 108)(72 109)(73 110)(74 106)(75 107)(76 127)(77 128)(78 129)(79 130)(80 126)(81 135)(82 131)(83 132)(84 133)(85 134)(86 120)(87 116)(88 117)(89 118)(90 119)(91 122)(92 123)(93 124)(94 125)(95 121)
(1 107 27 96)(2 108 28 97)(3 109 29 98)(4 110 30 99)(5 106 26 100)(6 94 14 85)(7 95 15 81)(8 91 11 82)(9 92 12 83)(10 93 13 84)(16 87 156 76)(17 88 157 77)(18 89 158 78)(19 90 159 79)(20 86 160 80)(21 115 35 101)(22 111 31 102)(23 112 32 103)(24 113 33 104)(25 114 34 105)(36 127 47 116)(37 128 48 117)(38 129 49 118)(39 130 50 119)(40 126 46 120)(41 135 55 121)(42 131 51 122)(43 132 52 123)(44 133 53 124)(45 134 54 125)(56 147 67 136)(57 148 68 137)(58 149 69 138)(59 150 70 139)(60 146 66 140)(61 155 75 141)(62 151 71 142)(63 152 72 143)(64 153 73 144)(65 154 74 145)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 134)(7 135)(8 131)(9 132)(10 133)(11 122)(12 123)(13 124)(14 125)(15 121)(16 127)(17 128)(18 129)(19 130)(20 126)(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)(96 136)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 140 160 154)(7 136 156 155)(8 137 157 151)(9 138 158 152)(10 139 159 153)(11 148 17 142)(12 149 18 143)(13 150 19 144)(14 146 20 145)(15 147 16 141)(21 47 27 41)(22 48 28 42)(23 49 29 43)(24 50 30 44)(25 46 26 45)(56 95 75 76)(57 91 71 77)(58 92 72 78)(59 93 73 79)(60 94 74 80)(61 87 67 81)(62 88 68 82)(63 89 69 83)(64 90 70 84)(65 86 66 85)(96 116 115 135)(97 117 111 131)(98 118 112 132)(99 119 113 133)(100 120 114 134)(101 121 107 127)(102 122 108 128)(103 123 109 129)(104 124 110 130)(105 125 106 126)
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)(6,160)(7,156)(8,157)(9,158)(10,159)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,115)(97,111)(98,112)(99,113)(100,114)(101,107)(102,108)(103,109)(104,110)(105,106)(116,135)(117,131)(118,132)(119,133)(120,134)(121,127)(122,128)(123,129)(124,130)(125,126)(136,155)(137,151)(138,152)(139,153)(140,154)(141,147)(142,148)(143,149)(144,150)(145,146), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73)(76,81)(77,82)(78,83)(79,84)(80,85)(86,94)(87,95)(88,91)(89,92)(90,93)(96,101)(97,102)(98,103)(99,104)(100,105)(106,114)(107,115)(108,111)(109,112)(110,113)(116,121)(117,122)(118,123)(119,124)(120,125)(126,134)(127,135)(128,131)(129,132)(130,133)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153), (1,136)(2,137)(3,138)(4,139)(5,140)(6,40)(7,36)(8,37)(9,38)(10,39)(11,48)(12,49)(13,50)(14,46)(15,47)(16,41)(17,42)(18,43)(19,44)(20,45)(21,141)(22,142)(23,143)(24,144)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(51,157)(52,158)(53,159)(54,160)(55,156)(56,101)(57,102)(58,103)(59,104)(60,105)(61,96)(62,97)(63,98)(64,99)(65,100)(66,114)(67,115)(68,111)(69,112)(70,113)(71,108)(72,109)(73,110)(74,106)(75,107)(76,127)(77,128)(78,129)(79,130)(80,126)(81,135)(82,131)(83,132)(84,133)(85,134)(86,120)(87,116)(88,117)(89,118)(90,119)(91,122)(92,123)(93,124)(94,125)(95,121), (1,107,27,96)(2,108,28,97)(3,109,29,98)(4,110,30,99)(5,106,26,100)(6,94,14,85)(7,95,15,81)(8,91,11,82)(9,92,12,83)(10,93,13,84)(16,87,156,76)(17,88,157,77)(18,89,158,78)(19,90,159,79)(20,86,160,80)(21,115,35,101)(22,111,31,102)(23,112,32,103)(24,113,33,104)(25,114,34,105)(36,127,47,116)(37,128,48,117)(38,129,49,118)(39,130,50,119)(40,126,46,120)(41,135,55,121)(42,131,51,122)(43,132,52,123)(44,133,53,124)(45,134,54,125)(56,147,67,136)(57,148,68,137)(58,149,69,138)(59,150,70,139)(60,146,66,140)(61,155,75,141)(62,151,71,142)(63,152,72,143)(64,153,73,144)(65,154,74,145), (1,56)(2,57)(3,58)(4,59)(5,60)(6,134)(7,135)(8,131)(9,132)(10,133)(11,122)(12,123)(13,124)(14,125)(15,121)(16,127)(17,128)(18,129)(19,130)(20,126)(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)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,140,160,154)(7,136,156,155)(8,137,157,151)(9,138,158,152)(10,139,159,153)(11,148,17,142)(12,149,18,143)(13,150,19,144)(14,146,20,145)(15,147,16,141)(21,47,27,41)(22,48,28,42)(23,49,29,43)(24,50,30,44)(25,46,26,45)(56,95,75,76)(57,91,71,77)(58,92,72,78)(59,93,73,79)(60,94,74,80)(61,87,67,81)(62,88,68,82)(63,89,69,83)(64,90,70,84)(65,86,66,85)(96,116,115,135)(97,117,111,131)(98,118,112,132)(99,119,113,133)(100,120,114,134)(101,121,107,127)(102,122,108,128)(103,123,109,129)(104,124,110,130)(105,125,106,126)>;
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)(6,160)(7,156)(8,157)(9,158)(10,159)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,115)(97,111)(98,112)(99,113)(100,114)(101,107)(102,108)(103,109)(104,110)(105,106)(116,135)(117,131)(118,132)(119,133)(120,134)(121,127)(122,128)(123,129)(124,130)(125,126)(136,155)(137,151)(138,152)(139,153)(140,154)(141,147)(142,148)(143,149)(144,150)(145,146), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73)(76,81)(77,82)(78,83)(79,84)(80,85)(86,94)(87,95)(88,91)(89,92)(90,93)(96,101)(97,102)(98,103)(99,104)(100,105)(106,114)(107,115)(108,111)(109,112)(110,113)(116,121)(117,122)(118,123)(119,124)(120,125)(126,134)(127,135)(128,131)(129,132)(130,133)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153), (1,136)(2,137)(3,138)(4,139)(5,140)(6,40)(7,36)(8,37)(9,38)(10,39)(11,48)(12,49)(13,50)(14,46)(15,47)(16,41)(17,42)(18,43)(19,44)(20,45)(21,141)(22,142)(23,143)(24,144)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(51,157)(52,158)(53,159)(54,160)(55,156)(56,101)(57,102)(58,103)(59,104)(60,105)(61,96)(62,97)(63,98)(64,99)(65,100)(66,114)(67,115)(68,111)(69,112)(70,113)(71,108)(72,109)(73,110)(74,106)(75,107)(76,127)(77,128)(78,129)(79,130)(80,126)(81,135)(82,131)(83,132)(84,133)(85,134)(86,120)(87,116)(88,117)(89,118)(90,119)(91,122)(92,123)(93,124)(94,125)(95,121), (1,107,27,96)(2,108,28,97)(3,109,29,98)(4,110,30,99)(5,106,26,100)(6,94,14,85)(7,95,15,81)(8,91,11,82)(9,92,12,83)(10,93,13,84)(16,87,156,76)(17,88,157,77)(18,89,158,78)(19,90,159,79)(20,86,160,80)(21,115,35,101)(22,111,31,102)(23,112,32,103)(24,113,33,104)(25,114,34,105)(36,127,47,116)(37,128,48,117)(38,129,49,118)(39,130,50,119)(40,126,46,120)(41,135,55,121)(42,131,51,122)(43,132,52,123)(44,133,53,124)(45,134,54,125)(56,147,67,136)(57,148,68,137)(58,149,69,138)(59,150,70,139)(60,146,66,140)(61,155,75,141)(62,151,71,142)(63,152,72,143)(64,153,73,144)(65,154,74,145), (1,56)(2,57)(3,58)(4,59)(5,60)(6,134)(7,135)(8,131)(9,132)(10,133)(11,122)(12,123)(13,124)(14,125)(15,121)(16,127)(17,128)(18,129)(19,130)(20,126)(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)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,140,160,154)(7,136,156,155)(8,137,157,151)(9,138,158,152)(10,139,159,153)(11,148,17,142)(12,149,18,143)(13,150,19,144)(14,146,20,145)(15,147,16,141)(21,47,27,41)(22,48,28,42)(23,49,29,43)(24,50,30,44)(25,46,26,45)(56,95,75,76)(57,91,71,77)(58,92,72,78)(59,93,73,79)(60,94,74,80)(61,87,67,81)(62,88,68,82)(63,89,69,83)(64,90,70,84)(65,86,66,85)(96,116,115,135)(97,117,111,131)(98,118,112,132)(99,119,113,133)(100,120,114,134)(101,121,107,127)(102,122,108,128)(103,123,109,129)(104,124,110,130)(105,125,106,126) );
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),(6,160),(7,156),(8,157),(9,158),(10,159),(11,17),(12,18),(13,19),(14,20),(15,16),(21,27),(22,28),(23,29),(24,30),(25,26),(36,55),(37,51),(38,52),(39,53),(40,54),(41,47),(42,48),(43,49),(44,50),(45,46),(56,75),(57,71),(58,72),(59,73),(60,74),(61,67),(62,68),(63,69),(64,70),(65,66),(76,95),(77,91),(78,92),(79,93),(80,94),(81,87),(82,88),(83,89),(84,90),(85,86),(96,115),(97,111),(98,112),(99,113),(100,114),(101,107),(102,108),(103,109),(104,110),(105,106),(116,135),(117,131),(118,132),(119,133),(120,134),(121,127),(122,128),(123,129),(124,130),(125,126),(136,155),(137,151),(138,152),(139,153),(140,154),(141,147),(142,148),(143,149),(144,150),(145,146)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,20),(7,16),(8,17),(9,18),(10,19),(11,157),(12,158),(13,159),(14,160),(15,156),(26,34),(27,35),(28,31),(29,32),(30,33),(36,41),(37,42),(38,43),(39,44),(40,45),(46,54),(47,55),(48,51),(49,52),(50,53),(56,61),(57,62),(58,63),(59,64),(60,65),(66,74),(67,75),(68,71),(69,72),(70,73),(76,81),(77,82),(78,83),(79,84),(80,85),(86,94),(87,95),(88,91),(89,92),(90,93),(96,101),(97,102),(98,103),(99,104),(100,105),(106,114),(107,115),(108,111),(109,112),(110,113),(116,121),(117,122),(118,123),(119,124),(120,125),(126,134),(127,135),(128,131),(129,132),(130,133),(136,141),(137,142),(138,143),(139,144),(140,145),(146,154),(147,155),(148,151),(149,152),(150,153)], [(1,136),(2,137),(3,138),(4,139),(5,140),(6,40),(7,36),(8,37),(9,38),(10,39),(11,48),(12,49),(13,50),(14,46),(15,47),(16,41),(17,42),(18,43),(19,44),(20,45),(21,141),(22,142),(23,143),(24,144),(25,145),(26,146),(27,147),(28,148),(29,149),(30,150),(31,151),(32,152),(33,153),(34,154),(35,155),(51,157),(52,158),(53,159),(54,160),(55,156),(56,101),(57,102),(58,103),(59,104),(60,105),(61,96),(62,97),(63,98),(64,99),(65,100),(66,114),(67,115),(68,111),(69,112),(70,113),(71,108),(72,109),(73,110),(74,106),(75,107),(76,127),(77,128),(78,129),(79,130),(80,126),(81,135),(82,131),(83,132),(84,133),(85,134),(86,120),(87,116),(88,117),(89,118),(90,119),(91,122),(92,123),(93,124),(94,125),(95,121)], [(1,107,27,96),(2,108,28,97),(3,109,29,98),(4,110,30,99),(5,106,26,100),(6,94,14,85),(7,95,15,81),(8,91,11,82),(9,92,12,83),(10,93,13,84),(16,87,156,76),(17,88,157,77),(18,89,158,78),(19,90,159,79),(20,86,160,80),(21,115,35,101),(22,111,31,102),(23,112,32,103),(24,113,33,104),(25,114,34,105),(36,127,47,116),(37,128,48,117),(38,129,49,118),(39,130,50,119),(40,126,46,120),(41,135,55,121),(42,131,51,122),(43,132,52,123),(44,133,53,124),(45,134,54,125),(56,147,67,136),(57,148,68,137),(58,149,69,138),(59,150,70,139),(60,146,66,140),(61,155,75,141),(62,151,71,142),(63,152,72,143),(64,153,73,144),(65,154,74,145)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,134),(7,135),(8,131),(9,132),(10,133),(11,122),(12,123),(13,124),(14,125),(15,121),(16,127),(17,128),(18,129),(19,130),(20,126),(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),(96,136),(97,137),(98,138),(99,139),(100,140),(101,141),(102,142),(103,143),(104,144),(105,145),(106,146),(107,147),(108,148),(109,149),(110,150),(111,151),(112,152),(113,153),(114,154),(115,155),(116,156),(117,157),(118,158),(119,159),(120,160)], [(1,55,35,36),(2,51,31,37),(3,52,32,38),(4,53,33,39),(5,54,34,40),(6,140,160,154),(7,136,156,155),(8,137,157,151),(9,138,158,152),(10,139,159,153),(11,148,17,142),(12,149,18,143),(13,150,19,144),(14,146,20,145),(15,147,16,141),(21,47,27,41),(22,48,28,42),(23,49,29,43),(24,50,30,44),(25,46,26,45),(56,95,75,76),(57,91,71,77),(58,92,72,78),(59,93,73,79),(60,94,74,80),(61,87,67,81),(62,88,68,82),(63,89,69,83),(64,90,70,84),(65,86,66,85),(96,116,115,135),(97,117,111,131),(98,118,112,132),(99,119,113,133),(100,120,114,134),(101,121,107,127),(102,122,108,128),(103,123,109,129),(104,124,110,130),(105,125,106,126)])
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4J | 4K | ··· | 4P | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AF | 20A | ··· | 20AN | 20AO | ··· | 20BL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | C4○D4 | C4○D4 | C5×C4○D4 | C5×C4○D4 | 2+ 1+4 | C5×2+ 1+4 |
| kernel | C5×C22.47C24 | C10×C4⋊C4 | C5×C42⋊C2 | D4×C20 | C5×C4⋊D4 | C5×C22.D4 | C5×C42.C2 | C5×C42⋊2C2 | C22.47C24 | C2×C4⋊C4 | C42⋊C2 | C4×D4 | C4⋊D4 | C22.D4 | C42.C2 | C42⋊2C2 | C20 | C2×C10 | C4 | C22 | C10 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 2 | 4 | 4 | 4 | 16 | 16 | 8 | 4 | 8 | 4 | 4 | 16 | 16 | 1 | 4 |
Matrix representation of C5×C22.47C24 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 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 | 32 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 32 | 18 |
| 0 | 0 | 32 | 9 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 18 |
| 0 | 0 | 0 | 9 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[0,9,0,0,32,0,0,0,0,0,32,32,0,0,18,9],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,18,9],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,1,0,0,39,40] >;
C5×C22.47C24 in GAP, Magma, Sage, TeX
C_5\times C_2^2._{47}C_2^4 % in TeX
G:=Group("C5xC2^2.47C2^4"); // GroupNames label
G:=SmallGroup(320,1555);
// by ID
G=gap.SmallGroup(320,1555);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1688,3446,1242,304]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=f^2=1,e^2=c*b=b*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,e*d*e^-1=b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
// generators/relations