direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C22.50C24, C10.1212- 1+4, C4⋊Q8⋊17C10, (Q8×C20)⋊36C2, (C4×Q8)⋊16C10, (D4×C20).28C2, (C4×D4).13C10, C22⋊Q8⋊18C10, C42⋊2C2⋊7C10, C42.50(C2×C10), C4.4D4.8C10, C42⋊C2⋊18C10, C20.347(C4○D4), (C2×C10).376C24, (C4×C20).291C22, (C2×C20).965C23, (D4×C10).325C22, C23.21(C22×C10), C22.50(C23×C10), (Q8×C10).277C22, C2.13(C5×2- 1+4), (C22×C10).104C23, (C22×C20).461C22, (C5×C4⋊Q8)⋊38C2, C4.39(C5×C4○D4), C4⋊C4.76(C2×C10), C2.29(C10×C4○D4), (C5×C22⋊Q8)⋊45C2, (C2×D4).71(C2×C10), C10.248(C2×C4○D4), C22⋊C4.6(C2×C10), (C2×Q8).65(C2×C10), (C5×C42⋊C2)⋊39C2, (C5×C42⋊2C2)⋊18C2, (C5×C4⋊C4).401C22, (C22×C4).72(C2×C10), (C2×C4).38(C22×C10), (C5×C4.4D4).17C2, (C5×C22⋊C4).90C22, SmallGroup(320,1558)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.50C24
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=1, e2=cb=bc, f2=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-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
Subgroups: 282 in 212 conjugacy classes, 150 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C20, C20, C2×C10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22.50C24, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, Q8×C10, C5×C42⋊C2, D4×C20, Q8×C20, Q8×C20, C5×C22⋊Q8, C5×C4.4D4, C5×C42⋊2C2, C5×C4⋊Q8, C5×C22.50C24
Quotients: C1, C2, C22, C5, C23, C10, C4○D4, C24, C2×C10, C2×C4○D4, 2- 1+4, C22×C10, C22.50C24, C5×C4○D4, C23×C10, C10×C4○D4, C5×2- 1+4, C5×C22.50C24
►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 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 156)(12 157)(13 158)(14 159)(15 160)(21 31)(22 32)(23 33)(24 34)(25 35)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)(76 86)(77 87)(78 88)(79 89)(80 90)(81 91)(82 92)(83 93)(84 94)(85 95)(96 106)(97 107)(98 108)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)(116 126)(117 127)(118 128)(119 129)(120 130)(121 131)(122 132)(123 133)(124 134)(125 135)(136 146)(137 147)(138 148)(139 149)(140 150)(141 151)(142 152)(143 153)(144 154)(145 155)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 156)(7 157)(8 158)(9 159)(10 160)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)(76 81)(77 82)(78 83)(79 84)(80 85)(86 91)(87 92)(88 93)(89 94)(90 95)(96 101)(97 102)(98 103)(99 104)(100 105)(106 111)(107 112)(108 113)(109 114)(110 115)(116 121)(117 122)(118 123)(119 124)(120 125)(126 131)(127 132)(128 133)(129 134)(130 135)(136 141)(137 142)(138 143)(139 144)(140 145)(146 151)(147 152)(148 153)(149 154)(150 155)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(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 71 31 56)(2 72 32 57)(3 73 33 58)(4 74 34 59)(5 75 35 60)(6 131 11 116)(7 132 12 117)(8 133 13 118)(9 134 14 119)(10 135 15 120)(16 121 156 126)(17 122 157 127)(18 123 158 128)(19 124 159 129)(20 125 160 130)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(36 91 51 76)(37 92 52 77)(38 93 53 78)(39 94 54 79)(40 95 55 80)(41 86 46 81)(42 87 47 82)(43 88 48 83)(44 89 49 84)(45 90 50 85)(96 141 111 146)(97 142 112 147)(98 143 113 148)(99 144 114 149)(100 145 115 150)(101 136 106 151)(102 137 107 152)(103 138 108 153)(104 139 109 154)(105 140 110 155)
(1 36 26 46)(2 37 27 47)(3 38 28 48)(4 39 29 49)(5 40 30 50)(6 146 16 136)(7 147 17 137)(8 148 18 138)(9 149 19 139)(10 150 20 140)(11 141 156 151)(12 142 157 152)(13 143 158 153)(14 144 159 154)(15 145 160 155)(21 41 31 51)(22 42 32 52)(23 43 33 53)(24 44 34 54)(25 45 35 55)(56 76 66 86)(57 77 67 87)(58 78 68 88)(59 79 69 89)(60 80 70 90)(61 81 71 91)(62 82 72 92)(63 83 73 93)(64 84 74 94)(65 85 75 95)(96 121 106 131)(97 122 107 132)(98 123 108 133)(99 124 109 134)(100 125 110 135)(101 116 111 126)(102 117 112 127)(103 118 113 128)(104 119 114 129)(105 120 115 130)
(1 46 26 36)(2 47 27 37)(3 48 28 38)(4 49 29 39)(5 50 30 40)(6 151 16 141)(7 152 17 142)(8 153 18 143)(9 154 19 144)(10 155 20 145)(11 136 156 146)(12 137 157 147)(13 138 158 148)(14 139 159 149)(15 140 160 150)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)(56 76 66 86)(57 77 67 87)(58 78 68 88)(59 79 69 89)(60 80 70 90)(61 81 71 91)(62 82 72 92)(63 83 73 93)(64 84 74 94)(65 85 75 95)(96 126 106 116)(97 127 107 117)(98 128 108 118)(99 129 109 119)(100 130 110 120)(101 131 111 121)(102 132 112 122)(103 133 113 123)(104 134 114 124)(105 135 115 125)
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), (1,26)(2,27)(3,28)(4,29)(5,30)(6,16)(7,17)(8,18)(9,19)(10,20)(11,156)(12,157)(13,158)(14,159)(15,160)(21,31)(22,32)(23,33)(24,34)(25,35)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75)(76,86)(77,87)(78,88)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94)(85,95)(96,106)(97,107)(98,108)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(116,126)(117,127)(118,128)(119,129)(120,130)(121,131)(122,132)(123,133)(124,134)(125,135)(136,146)(137,147)(138,148)(139,149)(140,150)(141,151)(142,152)(143,153)(144,154)(145,155), (1,21)(2,22)(3,23)(4,24)(5,25)(6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75)(76,81)(77,82)(78,83)(79,84)(80,85)(86,91)(87,92)(88,93)(89,94)(90,95)(96,101)(97,102)(98,103)(99,104)(100,105)(106,111)(107,112)(108,113)(109,114)(110,115)(116,121)(117,122)(118,123)(119,124)(120,125)(126,131)(127,132)(128,133)(129,134)(130,135)(136,141)(137,142)(138,143)(139,144)(140,145)(146,151)(147,152)(148,153)(149,154)(150,155), (1,96)(2,97)(3,98)(4,99)(5,100)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(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), (1,71,31,56)(2,72,32,57)(3,73,33,58)(4,74,34,59)(5,75,35,60)(6,131,11,116)(7,132,12,117)(8,133,13,118)(9,134,14,119)(10,135,15,120)(16,121,156,126)(17,122,157,127)(18,123,158,128)(19,124,159,129)(20,125,160,130)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(36,91,51,76)(37,92,52,77)(38,93,53,78)(39,94,54,79)(40,95,55,80)(41,86,46,81)(42,87,47,82)(43,88,48,83)(44,89,49,84)(45,90,50,85)(96,141,111,146)(97,142,112,147)(98,143,113,148)(99,144,114,149)(100,145,115,150)(101,136,106,151)(102,137,107,152)(103,138,108,153)(104,139,109,154)(105,140,110,155), (1,36,26,46)(2,37,27,47)(3,38,28,48)(4,39,29,49)(5,40,30,50)(6,146,16,136)(7,147,17,137)(8,148,18,138)(9,149,19,139)(10,150,20,140)(11,141,156,151)(12,142,157,152)(13,143,158,153)(14,144,159,154)(15,145,160,155)(21,41,31,51)(22,42,32,52)(23,43,33,53)(24,44,34,54)(25,45,35,55)(56,76,66,86)(57,77,67,87)(58,78,68,88)(59,79,69,89)(60,80,70,90)(61,81,71,91)(62,82,72,92)(63,83,73,93)(64,84,74,94)(65,85,75,95)(96,121,106,131)(97,122,107,132)(98,123,108,133)(99,124,109,134)(100,125,110,135)(101,116,111,126)(102,117,112,127)(103,118,113,128)(104,119,114,129)(105,120,115,130), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,151,16,141)(7,152,17,142)(8,153,18,143)(9,154,19,144)(10,155,20,145)(11,136,156,146)(12,137,157,147)(13,138,158,148)(14,139,159,149)(15,140,160,150)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45)(56,76,66,86)(57,77,67,87)(58,78,68,88)(59,79,69,89)(60,80,70,90)(61,81,71,91)(62,82,72,92)(63,83,73,93)(64,84,74,94)(65,85,75,95)(96,126,106,116)(97,127,107,117)(98,128,108,118)(99,129,109,119)(100,130,110,120)(101,131,111,121)(102,132,112,122)(103,133,113,123)(104,134,114,124)(105,135,115,125)>;
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), (1,26)(2,27)(3,28)(4,29)(5,30)(6,16)(7,17)(8,18)(9,19)(10,20)(11,156)(12,157)(13,158)(14,159)(15,160)(21,31)(22,32)(23,33)(24,34)(25,35)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75)(76,86)(77,87)(78,88)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94)(85,95)(96,106)(97,107)(98,108)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(116,126)(117,127)(118,128)(119,129)(120,130)(121,131)(122,132)(123,133)(124,134)(125,135)(136,146)(137,147)(138,148)(139,149)(140,150)(141,151)(142,152)(143,153)(144,154)(145,155), (1,21)(2,22)(3,23)(4,24)(5,25)(6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75)(76,81)(77,82)(78,83)(79,84)(80,85)(86,91)(87,92)(88,93)(89,94)(90,95)(96,101)(97,102)(98,103)(99,104)(100,105)(106,111)(107,112)(108,113)(109,114)(110,115)(116,121)(117,122)(118,123)(119,124)(120,125)(126,131)(127,132)(128,133)(129,134)(130,135)(136,141)(137,142)(138,143)(139,144)(140,145)(146,151)(147,152)(148,153)(149,154)(150,155), (1,96)(2,97)(3,98)(4,99)(5,100)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(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), (1,71,31,56)(2,72,32,57)(3,73,33,58)(4,74,34,59)(5,75,35,60)(6,131,11,116)(7,132,12,117)(8,133,13,118)(9,134,14,119)(10,135,15,120)(16,121,156,126)(17,122,157,127)(18,123,158,128)(19,124,159,129)(20,125,160,130)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(36,91,51,76)(37,92,52,77)(38,93,53,78)(39,94,54,79)(40,95,55,80)(41,86,46,81)(42,87,47,82)(43,88,48,83)(44,89,49,84)(45,90,50,85)(96,141,111,146)(97,142,112,147)(98,143,113,148)(99,144,114,149)(100,145,115,150)(101,136,106,151)(102,137,107,152)(103,138,108,153)(104,139,109,154)(105,140,110,155), (1,36,26,46)(2,37,27,47)(3,38,28,48)(4,39,29,49)(5,40,30,50)(6,146,16,136)(7,147,17,137)(8,148,18,138)(9,149,19,139)(10,150,20,140)(11,141,156,151)(12,142,157,152)(13,143,158,153)(14,144,159,154)(15,145,160,155)(21,41,31,51)(22,42,32,52)(23,43,33,53)(24,44,34,54)(25,45,35,55)(56,76,66,86)(57,77,67,87)(58,78,68,88)(59,79,69,89)(60,80,70,90)(61,81,71,91)(62,82,72,92)(63,83,73,93)(64,84,74,94)(65,85,75,95)(96,121,106,131)(97,122,107,132)(98,123,108,133)(99,124,109,134)(100,125,110,135)(101,116,111,126)(102,117,112,127)(103,118,113,128)(104,119,114,129)(105,120,115,130), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,151,16,141)(7,152,17,142)(8,153,18,143)(9,154,19,144)(10,155,20,145)(11,136,156,146)(12,137,157,147)(13,138,158,148)(14,139,159,149)(15,140,160,150)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45)(56,76,66,86)(57,77,67,87)(58,78,68,88)(59,79,69,89)(60,80,70,90)(61,81,71,91)(62,82,72,92)(63,83,73,93)(64,84,74,94)(65,85,75,95)(96,126,106,116)(97,127,107,117)(98,128,108,118)(99,129,109,119)(100,130,110,120)(101,131,111,121)(102,132,112,122)(103,133,113,123)(104,134,114,124)(105,135,115,125) );
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)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,16),(7,17),(8,18),(9,19),(10,20),(11,156),(12,157),(13,158),(14,159),(15,160),(21,31),(22,32),(23,33),(24,34),(25,35),(36,46),(37,47),(38,48),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(56,66),(57,67),(58,68),(59,69),(60,70),(61,71),(62,72),(63,73),(64,74),(65,75),(76,86),(77,87),(78,88),(79,89),(80,90),(81,91),(82,92),(83,93),(84,94),(85,95),(96,106),(97,107),(98,108),(99,109),(100,110),(101,111),(102,112),(103,113),(104,114),(105,115),(116,126),(117,127),(118,128),(119,129),(120,130),(121,131),(122,132),(123,133),(124,134),(125,135),(136,146),(137,147),(138,148),(139,149),(140,150),(141,151),(142,152),(143,153),(144,154),(145,155)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,156),(7,157),(8,158),(9,159),(10,160),(11,16),(12,17),(13,18),(14,19),(15,20),(26,31),(27,32),(28,33),(29,34),(30,35),(36,41),(37,42),(38,43),(39,44),(40,45),(46,51),(47,52),(48,53),(49,54),(50,55),(56,61),(57,62),(58,63),(59,64),(60,65),(66,71),(67,72),(68,73),(69,74),(70,75),(76,81),(77,82),(78,83),(79,84),(80,85),(86,91),(87,92),(88,93),(89,94),(90,95),(96,101),(97,102),(98,103),(99,104),(100,105),(106,111),(107,112),(108,113),(109,114),(110,115),(116,121),(117,122),(118,123),(119,124),(120,125),(126,131),(127,132),(128,133),(129,134),(130,135),(136,141),(137,142),(138,143),(139,144),(140,145),(146,151),(147,152),(148,153),(149,154),(150,155)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,81),(7,82),(8,83),(9,84),(10,85),(11,86),(12,87),(13,88),(14,89),(15,90),(16,91),(17,92),(18,93),(19,94),(20,95),(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)], [(1,71,31,56),(2,72,32,57),(3,73,33,58),(4,74,34,59),(5,75,35,60),(6,131,11,116),(7,132,12,117),(8,133,13,118),(9,134,14,119),(10,135,15,120),(16,121,156,126),(17,122,157,127),(18,123,158,128),(19,124,159,129),(20,125,160,130),(21,66,26,61),(22,67,27,62),(23,68,28,63),(24,69,29,64),(25,70,30,65),(36,91,51,76),(37,92,52,77),(38,93,53,78),(39,94,54,79),(40,95,55,80),(41,86,46,81),(42,87,47,82),(43,88,48,83),(44,89,49,84),(45,90,50,85),(96,141,111,146),(97,142,112,147),(98,143,113,148),(99,144,114,149),(100,145,115,150),(101,136,106,151),(102,137,107,152),(103,138,108,153),(104,139,109,154),(105,140,110,155)], [(1,36,26,46),(2,37,27,47),(3,38,28,48),(4,39,29,49),(5,40,30,50),(6,146,16,136),(7,147,17,137),(8,148,18,138),(9,149,19,139),(10,150,20,140),(11,141,156,151),(12,142,157,152),(13,143,158,153),(14,144,159,154),(15,145,160,155),(21,41,31,51),(22,42,32,52),(23,43,33,53),(24,44,34,54),(25,45,35,55),(56,76,66,86),(57,77,67,87),(58,78,68,88),(59,79,69,89),(60,80,70,90),(61,81,71,91),(62,82,72,92),(63,83,73,93),(64,84,74,94),(65,85,75,95),(96,121,106,131),(97,122,107,132),(98,123,108,133),(99,124,109,134),(100,125,110,135),(101,116,111,126),(102,117,112,127),(103,118,113,128),(104,119,114,129),(105,120,115,130)], [(1,46,26,36),(2,47,27,37),(3,48,28,38),(4,49,29,39),(5,50,30,40),(6,151,16,141),(7,152,17,142),(8,153,18,143),(9,154,19,144),(10,155,20,145),(11,136,156,146),(12,137,157,147),(13,138,158,148),(14,139,159,149),(15,140,160,150),(21,51,31,41),(22,52,32,42),(23,53,33,43),(24,54,34,44),(25,55,35,45),(56,76,66,86),(57,77,67,87),(58,78,68,88),(59,79,69,89),(60,80,70,90),(61,81,71,91),(62,82,72,92),(63,83,73,93),(64,84,74,94),(65,85,75,95),(96,126,106,116),(97,127,107,117),(98,128,108,118),(99,129,109,119),(100,130,110,120),(101,131,111,121),(102,132,112,122),(103,133,113,123),(104,134,114,124),(105,135,115,125)]])
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4S | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20AV | 20AW | ··· | 20BX |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | C4○D4 | C5×C4○D4 | 2- 1+4 | C5×2- 1+4 |
| kernel | C5×C22.50C24 | C5×C42⋊C2 | D4×C20 | Q8×C20 | C5×C22⋊Q8 | C5×C4.4D4 | C5×C42⋊2C2 | C5×C4⋊Q8 | C22.50C24 | C42⋊C2 | C4×D4 | C4×Q8 | C22⋊Q8 | C4.4D4 | C42⋊2C2 | C4⋊Q8 | C20 | C4 | C10 | C2 |
| # reps | 1 | 2 | 1 | 3 | 2 | 2 | 4 | 1 | 4 | 8 | 4 | 12 | 8 | 8 | 16 | 4 | 8 | 32 | 1 | 4 |
Matrix representation of C5×C22.50C24 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,40,0,0,1,0,0,0,0,0,32,0,0,0,0,32],[9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,40],[32,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1] >;
C5×C22.50C24 in GAP, Magma, Sage, TeX
C_5\times C_2^2._{50}C_2^4 % in TeX
G:=Group("C5xC2^2.50C2^4"); // GroupNames label
G:=SmallGroup(320,1558);
// by ID
G=gap.SmallGroup(320,1558);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1688,3446,436,1242,304]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=1,e^2=c*b=b*c,f^2=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^-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