direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C22.46C24, C10.1202- 1+4, (Q8×C20)⋊34C2, (C4×Q8)⋊14C10, (C4×D4).11C10, (D4×C20).26C2, C22⋊Q8⋊16C10, C42.C2⋊8C10, C42⋊2C2⋊5C10, C42.46(C2×C10), C42⋊C2⋊15C10, C20.323(C4○D4), (C2×C10).372C24, (C4×C20).287C22, (C2×C20).679C23, (D4×C10).324C22, C23.44(C22×C10), C22.46(C23×C10), (Q8×C10).276C22, C22.D4.2C10, C2.12(C5×2- 1+4), (C22×C10).267C23, (C22×C20).457C22, (C2×C4⋊C4)⋊21C10, (C10×C4⋊C4)⋊48C2, C4.35(C5×C4○D4), C4⋊C4.33(C2×C10), C2.25(C10×C4○D4), (C5×C22⋊Q8)⋊43C2, (C2×D4).70(C2×C10), C10.244(C2×C4○D4), (C2×Q8).63(C2×C10), (C5×C42.C2)⋊25C2, C22.10(C5×C4○D4), (C5×C42⋊2C2)⋊16C2, (C5×C42⋊C2)⋊36C2, (C5×C4⋊C4).250C22, C22⋊C4.22(C2×C10), (C22×C4).69(C2×C10), (C2×C4).35(C22×C10), (C2×C10).119(C4○D4), (C5×C22⋊C4).89C22, (C5×C22.D4).5C2, SmallGroup(320,1554)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.46C24
G = < a,b,c,d,e,f,g | a5=b2=c2=g2=1, d2=f2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
Subgroups: 282 in 214 conjugacy classes, 150 normal (62 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22.46C24, C4×C20, C4×C20, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, Q8×C10, C10×C4⋊C4, C5×C42⋊C2, C5×C42⋊C2, D4×C20, Q8×C20, C5×C22⋊Q8, C5×C22.D4, C5×C42.C2, C5×C42.C2, C5×C42⋊2C2, C5×C22.46C24
Quotients: C1, C2, C22, C5, C23, C10, C4○D4, C24, C2×C10, C2×C4○D4, 2- 1+4, C22×C10, C22.46C24, C5×C4○D4, C23×C10, C10×C4○D4, C5×2- 1+4, C5×C22.46C24
►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)
(1 156 35 7)(2 157 31 8)(3 158 32 9)(4 159 33 10)(5 160 34 6)(11 28 17 22)(12 29 18 23)(13 30 19 24)(14 26 20 25)(15 27 16 21)(36 155 55 136)(37 151 51 137)(38 152 52 138)(39 153 53 139)(40 154 54 140)(41 147 47 141)(42 148 48 142)(43 149 49 143)(44 150 50 144)(45 146 46 145)(56 127 75 121)(57 128 71 122)(58 129 72 123)(59 130 73 124)(60 126 74 125)(61 135 67 116)(62 131 68 117)(63 132 69 118)(64 133 70 119)(65 134 66 120)(76 101 95 107)(77 102 91 108)(78 103 92 109)(79 104 93 110)(80 105 94 106)(81 96 87 115)(82 97 88 111)(83 98 89 112)(84 99 90 113)(85 100 86 114)
(1 101 21 96)(2 102 22 97)(3 103 23 98)(4 104 24 99)(5 105 25 100)(6 94 20 86)(7 95 16 87)(8 91 17 88)(9 92 18 89)(10 93 19 90)(11 82 157 77)(12 83 158 78)(13 84 159 79)(14 85 160 80)(15 81 156 76)(26 114 34 106)(27 115 35 107)(28 111 31 108)(29 112 32 109)(30 113 33 110)(36 121 41 116)(37 122 42 117)(38 123 43 118)(39 124 44 119)(40 125 45 120)(46 134 54 126)(47 135 55 127)(48 131 51 128)(49 132 52 129)(50 133 53 130)(56 141 61 136)(57 142 62 137)(58 143 63 138)(59 144 64 139)(60 145 65 140)(66 154 74 146)(67 155 75 147)(68 151 71 148)(69 152 72 149)(70 153 73 150)
(1 75 35 56)(2 71 31 57)(3 72 32 58)(4 73 33 59)(5 74 34 60)(6 134 160 120)(7 135 156 116)(8 131 157 117)(9 132 158 118)(10 133 159 119)(11 122 17 128)(12 123 18 129)(13 124 19 130)(14 125 20 126)(15 121 16 127)(21 67 27 61)(22 68 28 62)(23 69 29 63)(24 70 30 64)(25 66 26 65)(36 95 55 76)(37 91 51 77)(38 92 52 78)(39 93 53 79)(40 94 54 80)(41 87 47 81)(42 88 48 82)(43 89 49 83)(44 90 50 84)(45 86 46 85)(96 155 115 136)(97 151 111 137)(98 152 112 138)(99 153 113 139)(100 154 114 140)(101 147 107 141)(102 148 108 142)(103 149 109 143)(104 150 110 144)(105 146 106 145)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 140)(7 136)(8 137)(9 138)(10 139)(11 148)(12 149)(13 150)(14 146)(15 147)(16 141)(17 142)(18 143)(19 144)(20 145)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(56 76)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(96 135)(97 131)(98 132)(99 133)(100 134)(101 127)(102 128)(103 129)(104 130)(105 126)(106 125)(107 121)(108 122)(109 123)(110 124)(111 117)(112 118)(113 119)(114 120)(115 116)(151 157)(152 158)(153 159)(154 160)(155 156)
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), (1,156,35,7)(2,157,31,8)(3,158,32,9)(4,159,33,10)(5,160,34,6)(11,28,17,22)(12,29,18,23)(13,30,19,24)(14,26,20,25)(15,27,16,21)(36,155,55,136)(37,151,51,137)(38,152,52,138)(39,153,53,139)(40,154,54,140)(41,147,47,141)(42,148,48,142)(43,149,49,143)(44,150,50,144)(45,146,46,145)(56,127,75,121)(57,128,71,122)(58,129,72,123)(59,130,73,124)(60,126,74,125)(61,135,67,116)(62,131,68,117)(63,132,69,118)(64,133,70,119)(65,134,66,120)(76,101,95,107)(77,102,91,108)(78,103,92,109)(79,104,93,110)(80,105,94,106)(81,96,87,115)(82,97,88,111)(83,98,89,112)(84,99,90,113)(85,100,86,114), (1,101,21,96)(2,102,22,97)(3,103,23,98)(4,104,24,99)(5,105,25,100)(6,94,20,86)(7,95,16,87)(8,91,17,88)(9,92,18,89)(10,93,19,90)(11,82,157,77)(12,83,158,78)(13,84,159,79)(14,85,160,80)(15,81,156,76)(26,114,34,106)(27,115,35,107)(28,111,31,108)(29,112,32,109)(30,113,33,110)(36,121,41,116)(37,122,42,117)(38,123,43,118)(39,124,44,119)(40,125,45,120)(46,134,54,126)(47,135,55,127)(48,131,51,128)(49,132,52,129)(50,133,53,130)(56,141,61,136)(57,142,62,137)(58,143,63,138)(59,144,64,139)(60,145,65,140)(66,154,74,146)(67,155,75,147)(68,151,71,148)(69,152,72,149)(70,153,73,150), (1,75,35,56)(2,71,31,57)(3,72,32,58)(4,73,33,59)(5,74,34,60)(6,134,160,120)(7,135,156,116)(8,131,157,117)(9,132,158,118)(10,133,159,119)(11,122,17,128)(12,123,18,129)(13,124,19,130)(14,125,20,126)(15,121,16,127)(21,67,27,61)(22,68,28,62)(23,69,29,63)(24,70,30,64)(25,66,26,65)(36,95,55,76)(37,91,51,77)(38,92,52,78)(39,93,53,79)(40,94,54,80)(41,87,47,81)(42,88,48,82)(43,89,49,83)(44,90,50,84)(45,86,46,85)(96,155,115,136)(97,151,111,137)(98,152,112,138)(99,153,113,139)(100,154,114,140)(101,147,107,141)(102,148,108,142)(103,149,109,143)(104,150,110,144)(105,146,106,145), (1,36)(2,37)(3,38)(4,39)(5,40)(6,140)(7,136)(8,137)(9,138)(10,139)(11,148)(12,149)(13,150)(14,146)(15,147)(16,141)(17,142)(18,143)(19,144)(20,145)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(96,135)(97,131)(98,132)(99,133)(100,134)(101,127)(102,128)(103,129)(104,130)(105,126)(106,125)(107,121)(108,122)(109,123)(110,124)(111,117)(112,118)(113,119)(114,120)(115,116)(151,157)(152,158)(153,159)(154,160)(155,156)>;
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), (1,156,35,7)(2,157,31,8)(3,158,32,9)(4,159,33,10)(5,160,34,6)(11,28,17,22)(12,29,18,23)(13,30,19,24)(14,26,20,25)(15,27,16,21)(36,155,55,136)(37,151,51,137)(38,152,52,138)(39,153,53,139)(40,154,54,140)(41,147,47,141)(42,148,48,142)(43,149,49,143)(44,150,50,144)(45,146,46,145)(56,127,75,121)(57,128,71,122)(58,129,72,123)(59,130,73,124)(60,126,74,125)(61,135,67,116)(62,131,68,117)(63,132,69,118)(64,133,70,119)(65,134,66,120)(76,101,95,107)(77,102,91,108)(78,103,92,109)(79,104,93,110)(80,105,94,106)(81,96,87,115)(82,97,88,111)(83,98,89,112)(84,99,90,113)(85,100,86,114), (1,101,21,96)(2,102,22,97)(3,103,23,98)(4,104,24,99)(5,105,25,100)(6,94,20,86)(7,95,16,87)(8,91,17,88)(9,92,18,89)(10,93,19,90)(11,82,157,77)(12,83,158,78)(13,84,159,79)(14,85,160,80)(15,81,156,76)(26,114,34,106)(27,115,35,107)(28,111,31,108)(29,112,32,109)(30,113,33,110)(36,121,41,116)(37,122,42,117)(38,123,43,118)(39,124,44,119)(40,125,45,120)(46,134,54,126)(47,135,55,127)(48,131,51,128)(49,132,52,129)(50,133,53,130)(56,141,61,136)(57,142,62,137)(58,143,63,138)(59,144,64,139)(60,145,65,140)(66,154,74,146)(67,155,75,147)(68,151,71,148)(69,152,72,149)(70,153,73,150), (1,75,35,56)(2,71,31,57)(3,72,32,58)(4,73,33,59)(5,74,34,60)(6,134,160,120)(7,135,156,116)(8,131,157,117)(9,132,158,118)(10,133,159,119)(11,122,17,128)(12,123,18,129)(13,124,19,130)(14,125,20,126)(15,121,16,127)(21,67,27,61)(22,68,28,62)(23,69,29,63)(24,70,30,64)(25,66,26,65)(36,95,55,76)(37,91,51,77)(38,92,52,78)(39,93,53,79)(40,94,54,80)(41,87,47,81)(42,88,48,82)(43,89,49,83)(44,90,50,84)(45,86,46,85)(96,155,115,136)(97,151,111,137)(98,152,112,138)(99,153,113,139)(100,154,114,140)(101,147,107,141)(102,148,108,142)(103,149,109,143)(104,150,110,144)(105,146,106,145), (1,36)(2,37)(3,38)(4,39)(5,40)(6,140)(7,136)(8,137)(9,138)(10,139)(11,148)(12,149)(13,150)(14,146)(15,147)(16,141)(17,142)(18,143)(19,144)(20,145)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(96,135)(97,131)(98,132)(99,133)(100,134)(101,127)(102,128)(103,129)(104,130)(105,126)(106,125)(107,121)(108,122)(109,123)(110,124)(111,117)(112,118)(113,119)(114,120)(115,116)(151,157)(152,158)(153,159)(154,160)(155,156) );
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)], [(1,156,35,7),(2,157,31,8),(3,158,32,9),(4,159,33,10),(5,160,34,6),(11,28,17,22),(12,29,18,23),(13,30,19,24),(14,26,20,25),(15,27,16,21),(36,155,55,136),(37,151,51,137),(38,152,52,138),(39,153,53,139),(40,154,54,140),(41,147,47,141),(42,148,48,142),(43,149,49,143),(44,150,50,144),(45,146,46,145),(56,127,75,121),(57,128,71,122),(58,129,72,123),(59,130,73,124),(60,126,74,125),(61,135,67,116),(62,131,68,117),(63,132,69,118),(64,133,70,119),(65,134,66,120),(76,101,95,107),(77,102,91,108),(78,103,92,109),(79,104,93,110),(80,105,94,106),(81,96,87,115),(82,97,88,111),(83,98,89,112),(84,99,90,113),(85,100,86,114)], [(1,101,21,96),(2,102,22,97),(3,103,23,98),(4,104,24,99),(5,105,25,100),(6,94,20,86),(7,95,16,87),(8,91,17,88),(9,92,18,89),(10,93,19,90),(11,82,157,77),(12,83,158,78),(13,84,159,79),(14,85,160,80),(15,81,156,76),(26,114,34,106),(27,115,35,107),(28,111,31,108),(29,112,32,109),(30,113,33,110),(36,121,41,116),(37,122,42,117),(38,123,43,118),(39,124,44,119),(40,125,45,120),(46,134,54,126),(47,135,55,127),(48,131,51,128),(49,132,52,129),(50,133,53,130),(56,141,61,136),(57,142,62,137),(58,143,63,138),(59,144,64,139),(60,145,65,140),(66,154,74,146),(67,155,75,147),(68,151,71,148),(69,152,72,149),(70,153,73,150)], [(1,75,35,56),(2,71,31,57),(3,72,32,58),(4,73,33,59),(5,74,34,60),(6,134,160,120),(7,135,156,116),(8,131,157,117),(9,132,158,118),(10,133,159,119),(11,122,17,128),(12,123,18,129),(13,124,19,130),(14,125,20,126),(15,121,16,127),(21,67,27,61),(22,68,28,62),(23,69,29,63),(24,70,30,64),(25,66,26,65),(36,95,55,76),(37,91,51,77),(38,92,52,78),(39,93,53,79),(40,94,54,80),(41,87,47,81),(42,88,48,82),(43,89,49,83),(44,90,50,84),(45,86,46,85),(96,155,115,136),(97,151,111,137),(98,152,112,138),(99,153,113,139),(100,154,114,140),(101,147,107,141),(102,148,108,142),(103,149,109,143),(104,150,110,144),(105,146,106,145)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,140),(7,136),(8,137),(9,138),(10,139),(11,148),(12,149),(13,150),(14,146),(15,147),(16,141),(17,142),(18,143),(19,144),(20,145),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(56,76),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,91),(72,92),(73,93),(74,94),(75,95),(96,135),(97,131),(98,132),(99,133),(100,134),(101,127),(102,128),(103,129),(104,130),(105,126),(106,125),(107,121),(108,122),(109,123),(110,124),(111,117),(112,118),(113,119),(114,120),(115,116),(151,157),(152,158),(153,159),(154,160),(155,156)]])
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4J | 4K | ··· | 4R | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | 10V | 10W | 10X | 20A | ··· | 20AN | 20AO | ··· | 20BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | 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×C22.46C24 | C10×C4⋊C4 | C5×C42⋊C2 | D4×C20 | Q8×C20 | C5×C22⋊Q8 | C5×C22.D4 | C5×C42.C2 | C5×C42⋊2C2 | C22.46C24 | C2×C4⋊C4 | C42⋊C2 | C4×D4 | C4×Q8 | C22⋊Q8 | C22.D4 | C42.C2 | C42⋊2C2 | C20 | C2×C10 | C4 | C22 | C10 | C2 |
| # reps | 1 | 1 | 3 | 1 | 1 | 2 | 2 | 3 | 2 | 4 | 4 | 12 | 4 | 4 | 8 | 8 | 12 | 8 | 4 | 4 | 16 | 16 | 1 | 4 |
Matrix representation of C5×C22.46C24 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 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 | 36 | 23 |
| 0 | 0 | 6 | 5 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 4 | 40 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 39 |
| 0 | 0 | 28 | 37 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[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],[0,32,0,0,9,0,0,0,0,0,36,6,0,0,23,5],[9,0,0,0,0,9,0,0,0,0,1,4,0,0,0,40],[0,1,0,0,1,0,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,4,28,0,0,39,37] >;
C5×C22.46C24 in GAP, Magma, Sage, TeX
C_5\times C_2^2._{46}C_2^4 % in TeX
G:=Group("C5xC2^2.46C2^4"); // GroupNames label
G:=SmallGroup(320,1554);
// by ID
G=gap.SmallGroup(320,1554);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,1128,3446,436,1242]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=g^2=1,d^2=f^2=b,e^2=c,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^-1=b*d=d*b,g*e*g=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,f*g=g*f>;
// generators/relations