direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C22.31C24, C10.1122- (1+4), C10.1532+ (1+4), (C2×C20)⋊27D4, C4⋊D4⋊7C10, C4.65(D4×C10), C22⋊Q8⋊6C10, C20.472(C2×D4), C22.3(D4×C10), (C2×C10).357C24, (C2×C20).666C23, C10.192(C22×D4), C2.5(C5×2+ (1+4)), C2.4(C5×2- (1+4)), (D4×C10).216C22, (C22×C10).92C23, C22.31(C23×C10), C23.37(C22×C10), (Q8×C10).271C22, (C22×C20).447C22, (C2×C4)⋊5(C5×D4), (C10×C4⋊C4)⋊45C2, (C2×C4⋊C4)⋊18C10, C2.16(D4×C2×C10), (C2×C4○D4)⋊5C10, (C10×C4○D4)⋊21C2, C4⋊C4.28(C2×C10), (C5×C4⋊D4)⋊34C2, (C2×C10).91(C2×D4), (C5×C22⋊Q8)⋊33C2, (C2×D4).30(C2×C10), C22⋊C4.2(C2×C10), (C2×Q8).58(C2×C10), (C5×C4⋊C4).391C22, (C22×C4).58(C2×C10), (C2×C4).24(C22×C10), (C5×C22⋊C4).84C22, SmallGroup(320,1539)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 466 in 294 conjugacy classes, 162 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×14], C5, C2×C4 [×14], C2×C4 [×10], D4 [×16], Q8 [×4], C23, C23 [×4], C10 [×3], C10 [×6], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C20 [×4], C20 [×8], C2×C10, C2×C10 [×2], C2×C10 [×14], C2×C4⋊C4, C4⋊D4 [×8], C22⋊Q8 [×4], C2×C4○D4 [×2], C2×C20 [×14], C2×C20 [×10], C5×D4 [×16], C5×Q8 [×4], C22×C10, C22×C10 [×4], C22.31C24, C5×C22⋊C4 [×8], C5×C4⋊C4 [×8], C22×C20, C22×C20 [×6], D4×C10 [×10], Q8×C10 [×2], C5×C4○D4 [×8], C10×C4⋊C4, C5×C4⋊D4 [×8], C5×C22⋊Q8 [×4], C10×C4○D4 [×2], C5×C22.31C24
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C24, C2×C10 [×35], C22×D4, 2+ (1+4), 2- (1+4), C5×D4 [×4], C22×C10 [×15], C22.31C24, D4×C10 [×6], C23×C10, D4×C2×C10, C5×2+ (1+4), C5×2- (1+4), C5×C22.31C24
Generators and relations
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=e2=f2=1, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >
►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 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 76)(2 77)(3 78)(4 79)(5 80)(6 103)(7 104)(8 105)(9 101)(10 102)(11 96)(12 97)(13 98)(14 99)(15 100)(16 110)(17 106)(18 107)(19 108)(20 109)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 75)(37 71)(38 72)(39 73)(40 74)(41 68)(42 69)(43 70)(44 66)(45 67)(46 64)(47 65)(48 61)(49 62)(50 63)(51 57)(52 58)(53 59)(54 60)(55 56)(111 157)(112 158)(113 159)(114 160)(115 156)(116 136)(117 137)(118 138)(119 139)(120 140)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 116)(7 117)(8 118)(9 119)(10 120)(11 123)(12 124)(13 125)(14 121)(15 122)(16 135)(17 131)(18 132)(19 133)(20 134)(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)(96 150)(97 146)(98 147)(99 148)(100 149)(101 153)(102 154)(103 155)(104 151)(105 152)(106 137)(107 138)(108 139)(109 140)(110 136)(111 144)(112 145)(113 141)(114 142)(115 143)(126 157)(127 158)(128 159)(129 160)(130 156)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 143 16 150)(7 144 17 146)(8 145 18 147)(9 141 19 148)(10 142 20 149)(11 136 156 155)(12 137 157 151)(13 138 158 152)(14 139 159 153)(15 140 160 154)(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)(96 116 115 135)(97 117 111 131)(98 118 112 132)(99 119 113 133)(100 120 114 134)(101 121 108 128)(102 122 109 129)(103 123 110 130)(104 124 106 126)(105 125 107 127)
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), (1,96)(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), (1,76)(2,77)(3,78)(4,79)(5,80)(6,103)(7,104)(8,105)(9,101)(10,102)(11,96)(12,97)(13,98)(14,99)(15,100)(16,110)(17,106)(18,107)(19,108)(20,109)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,75)(37,71)(38,72)(39,73)(40,74)(41,68)(42,69)(43,70)(44,66)(45,67)(46,64)(47,65)(48,61)(49,62)(50,63)(51,57)(52,58)(53,59)(54,60)(55,56)(111,157)(112,158)(113,159)(114,160)(115,156)(116,136)(117,137)(118,138)(119,139)(120,140)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155), (1,56)(2,57)(3,58)(4,59)(5,60)(6,116)(7,117)(8,118)(9,119)(10,120)(11,123)(12,124)(13,125)(14,121)(15,122)(16,135)(17,131)(18,132)(19,133)(20,134)(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)(96,150)(97,146)(98,147)(99,148)(100,149)(101,153)(102,154)(103,155)(104,151)(105,152)(106,137)(107,138)(108,139)(109,140)(110,136)(111,144)(112,145)(113,141)(114,142)(115,143)(126,157)(127,158)(128,159)(129,160)(130,156), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,143,16,150)(7,144,17,146)(8,145,18,147)(9,141,19,148)(10,142,20,149)(11,136,156,155)(12,137,157,151)(13,138,158,152)(14,139,159,153)(15,140,160,154)(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)(96,116,115,135)(97,117,111,131)(98,118,112,132)(99,119,113,133)(100,120,114,134)(101,121,108,128)(102,122,109,129)(103,123,110,130)(104,124,106,126)(105,125,107,127)>;
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), (1,96)(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), (1,76)(2,77)(3,78)(4,79)(5,80)(6,103)(7,104)(8,105)(9,101)(10,102)(11,96)(12,97)(13,98)(14,99)(15,100)(16,110)(17,106)(18,107)(19,108)(20,109)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,75)(37,71)(38,72)(39,73)(40,74)(41,68)(42,69)(43,70)(44,66)(45,67)(46,64)(47,65)(48,61)(49,62)(50,63)(51,57)(52,58)(53,59)(54,60)(55,56)(111,157)(112,158)(113,159)(114,160)(115,156)(116,136)(117,137)(118,138)(119,139)(120,140)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155), (1,56)(2,57)(3,58)(4,59)(5,60)(6,116)(7,117)(8,118)(9,119)(10,120)(11,123)(12,124)(13,125)(14,121)(15,122)(16,135)(17,131)(18,132)(19,133)(20,134)(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)(96,150)(97,146)(98,147)(99,148)(100,149)(101,153)(102,154)(103,155)(104,151)(105,152)(106,137)(107,138)(108,139)(109,140)(110,136)(111,144)(112,145)(113,141)(114,142)(115,143)(126,157)(127,158)(128,159)(129,160)(130,156), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,143,16,150)(7,144,17,146)(8,145,18,147)(9,141,19,148)(10,142,20,149)(11,136,156,155)(12,137,157,151)(13,138,158,152)(14,139,159,153)(15,140,160,154)(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)(96,116,115,135)(97,117,111,131)(98,118,112,132)(99,119,113,133)(100,120,114,134)(101,121,108,128)(102,122,109,129)(103,123,110,130)(104,124,106,126)(105,125,107,127) );
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)], [(1,96),(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)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,103),(7,104),(8,105),(9,101),(10,102),(11,96),(12,97),(13,98),(14,99),(15,100),(16,110),(17,106),(18,107),(19,108),(20,109),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,75),(37,71),(38,72),(39,73),(40,74),(41,68),(42,69),(43,70),(44,66),(45,67),(46,64),(47,65),(48,61),(49,62),(50,63),(51,57),(52,58),(53,59),(54,60),(55,56),(111,157),(112,158),(113,159),(114,160),(115,156),(116,136),(117,137),(118,138),(119,139),(120,140),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146),(127,147),(128,148),(129,149),(130,150),(131,151),(132,152),(133,153),(134,154),(135,155)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,116),(7,117),(8,118),(9,119),(10,120),(11,123),(12,124),(13,125),(14,121),(15,122),(16,135),(17,131),(18,132),(19,133),(20,134),(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),(96,150),(97,146),(98,147),(99,148),(100,149),(101,153),(102,154),(103,155),(104,151),(105,152),(106,137),(107,138),(108,139),(109,140),(110,136),(111,144),(112,145),(113,141),(114,142),(115,143),(126,157),(127,158),(128,159),(129,160),(130,156)], [(1,55,35,36),(2,51,31,37),(3,52,32,38),(4,53,33,39),(5,54,34,40),(6,143,16,150),(7,144,17,146),(8,145,18,147),(9,141,19,148),(10,142,20,149),(11,136,156,155),(12,137,157,151),(13,138,158,152),(14,139,159,153),(15,140,160,154),(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),(96,116,115,135),(97,117,111,131),(98,118,112,132),(99,119,113,133),(100,120,114,134),(101,121,108,128),(102,122,109,129),(103,123,110,130),(104,124,106,126),(105,125,107,127)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 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 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 39 | 0 | 0 | 0 | 0 |
| 25 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 18 | 0 | 0 |
| 0 | 0 | 15 | 10 | 0 | 0 |
| 0 | 0 | 0 | 23 | 31 | 36 |
| 0 | 0 | 28 | 0 | 28 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 40 | 39 |
| 0 | 0 | 0 | 40 | 2 | 2 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 40 | 40 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 | 0 | 2 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 40 | 39 |
| 0 | 0 | 1 | 0 | 1 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,25,0,0,0,0,39,25,0,0,0,0,0,0,31,15,0,28,0,0,18,10,23,0,0,0,0,0,31,28,0,0,0,0,36,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,40,0,0,1,40,1,40,0,0,40,2,0,1,0,0,39,2,0,1],[1,16,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,2,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,2,0,1,0,0,40,1,1,0,0,0,0,0,40,1,0,0,0,0,39,1] >;
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AJ | 20A | ··· | 20P | 20Q | ··· | 20AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | D4 | C5×D4 | 2+ (1+4) | 2- (1+4) | C5×2+ (1+4) | C5×2- (1+4) |
| kernel | C5×C22.31C24 | C10×C4⋊C4 | C5×C4⋊D4 | C5×C22⋊Q8 | C10×C4○D4 | C22.31C24 | C2×C4⋊C4 | C4⋊D4 | C22⋊Q8 | C2×C4○D4 | C2×C20 | C2×C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 8 | 4 | 2 | 4 | 4 | 32 | 16 | 8 | 4 | 16 | 1 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_5\times C_2^2._{31}C_2^4 % in TeX
G:=Group("C5xC2^2.31C2^4"); // GroupNames label
G:=SmallGroup(320,1539);
// by ID
G=gap.SmallGroup(320,1539);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,891,2467,304]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=e^2=f^2=1,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=g*d*g^-1=b*d=d*b,f*e*f=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,e*g=g*e,f*g=g*f>;
// generators/relations
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