direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C23.37C23, C4⋊Q8⋊20C10, (C4×Q8)⋊8C10, (C2×C20)⋊16Q8, (Q8×C20)⋊28C2, C20.99(C2×Q8), C4.10(Q8×C10), C22.3(Q8×C10), C42.35(C2×C10), (C2×C42).20C10, C42.C2⋊14C10, C22⋊Q8.10C10, C20.275(C4○D4), C10.60(C22×Q8), (C2×C10).353C24, (C4×C20).376C22, (C2×C20).662C23, C42⋊C2.11C10, C22.27(C23×C10), C23.35(C22×C10), (Q8×C10).269C22, (C22×C10).260C23, (C22×C20).598C22, (C2×C4)⋊5(C5×Q8), C2.6(Q8×C2×C10), (C2×C4×C20).43C2, (C5×C4⋊Q8)⋊41C2, C4.19(C5×C4○D4), C4⋊C4.67(C2×C10), C2.14(C10×C4○D4), (C2×C10).16(C2×Q8), C10.233(C2×C4○D4), (C2×Q8).56(C2×C10), (C5×C42.C2)⋊31C2, (C5×C22⋊Q8).20C2, C22⋊C4.14(C2×C10), (C5×C4⋊C4).390C22, (C2×C4).20(C22×C10), (C5×C42⋊C2).25C2, (C22×C4).125(C2×C10), (C5×C22⋊C4).148C22, SmallGroup(320,1535)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.37C23
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=1, e2=f2=d, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe-1=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, eg=ge, fg=gf >
Subgroups: 274 in 222 conjugacy classes, 170 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×C20, C2×C20, C2×C20, C5×Q8, C22×C10, C23.37C23, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, Q8×C10, C2×C4×C20, C5×C42⋊C2, Q8×C20, C5×C22⋊Q8, C5×C42.C2, C5×C4⋊Q8, C5×C23.37C23
Quotients: C1, C2, C22, C5, Q8, C23, C10, C2×Q8, C4○D4, C24, C2×C10, C22×Q8, C2×C4○D4, C5×Q8, C22×C10, C23.37C23, Q8×C10, C5×C4○D4, C23×C10, Q8×C2×C10, C10×C4○D4, C5×C23.37C23
►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 21)(2 22)(3 23)(4 24)(5 25)(6 11)(7 12)(8 13)(9 14)(10 15)(16 156)(17 157)(18 158)(19 159)(20 160)(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)(96 111)(97 112)(98 113)(99 114)(100 115)(101 106)(102 107)(103 108)(104 109)(105 110)(116 131)(117 132)(118 133)(119 134)(120 135)(121 126)(122 127)(123 128)(124 129)(125 130)(136 151)(137 152)(138 153)(139 154)(140 155)(141 146)(142 147)(143 148)(144 149)(145 150)
(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)
(1 101 21 96)(2 102 22 97)(3 103 23 98)(4 104 24 99)(5 105 25 100)(6 81 156 76)(7 82 157 77)(8 83 158 78)(9 84 159 79)(10 85 160 80)(11 86 16 91)(12 87 17 92)(13 88 18 93)(14 89 19 94)(15 90 20 95)(26 111 31 106)(27 112 32 107)(28 113 33 108)(29 114 34 109)(30 115 35 110)(36 121 41 116)(37 122 42 117)(38 123 43 118)(39 124 44 119)(40 125 45 120)(46 131 51 126)(47 132 52 127)(48 133 53 128)(49 134 54 129)(50 135 55 130)(56 141 61 136)(57 142 62 137)(58 143 63 138)(59 144 64 139)(60 145 65 140)(66 151 71 146)(67 152 72 147)(68 153 73 148)(69 154 74 149)(70 155 75 150)
(1 61 21 56)(2 62 22 57)(3 63 23 58)(4 64 24 59)(5 65 25 60)(6 116 156 121)(7 117 157 122)(8 118 158 123)(9 119 159 124)(10 120 160 125)(11 131 16 126)(12 132 17 127)(13 133 18 128)(14 134 19 129)(15 135 20 130)(26 71 31 66)(27 72 32 67)(28 73 33 68)(29 74 34 69)(30 75 35 70)(36 81 41 76)(37 82 42 77)(38 83 43 78)(39 84 44 79)(40 85 45 80)(46 91 51 86)(47 92 52 87)(48 93 53 88)(49 94 54 89)(50 95 55 90)(96 136 101 141)(97 137 102 142)(98 138 103 143)(99 139 104 144)(100 140 105 145)(106 146 111 151)(107 147 112 152)(108 148 113 153)(109 149 114 154)(110 150 115 155)
(1 46 26 36)(2 47 27 37)(3 48 28 38)(4 49 29 39)(5 50 30 40)(6 141 16 151)(7 142 17 152)(8 143 18 153)(9 144 19 154)(10 145 20 155)(11 146 156 136)(12 147 157 137)(13 148 158 138)(14 149 159 139)(15 150 160 140)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)(56 86 66 76)(57 87 67 77)(58 88 68 78)(59 89 69 79)(60 90 70 80)(61 91 71 81)(62 92 72 82)(63 93 73 83)(64 94 74 84)(65 95 75 85)(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), (1,21)(2,22)(3,23)(4,24)(5,25)(6,11)(7,12)(8,13)(9,14)(10,15)(16,156)(17,157)(18,158)(19,159)(20,160)(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)(96,111)(97,112)(98,113)(99,114)(100,115)(101,106)(102,107)(103,108)(104,109)(105,110)(116,131)(117,132)(118,133)(119,134)(120,135)(121,126)(122,127)(123,128)(124,129)(125,130)(136,151)(137,152)(138,153)(139,154)(140,155)(141,146)(142,147)(143,148)(144,149)(145,150), (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), (1,101,21,96)(2,102,22,97)(3,103,23,98)(4,104,24,99)(5,105,25,100)(6,81,156,76)(7,82,157,77)(8,83,158,78)(9,84,159,79)(10,85,160,80)(11,86,16,91)(12,87,17,92)(13,88,18,93)(14,89,19,94)(15,90,20,95)(26,111,31,106)(27,112,32,107)(28,113,33,108)(29,114,34,109)(30,115,35,110)(36,121,41,116)(37,122,42,117)(38,123,43,118)(39,124,44,119)(40,125,45,120)(46,131,51,126)(47,132,52,127)(48,133,53,128)(49,134,54,129)(50,135,55,130)(56,141,61,136)(57,142,62,137)(58,143,63,138)(59,144,64,139)(60,145,65,140)(66,151,71,146)(67,152,72,147)(68,153,73,148)(69,154,74,149)(70,155,75,150), (1,61,21,56)(2,62,22,57)(3,63,23,58)(4,64,24,59)(5,65,25,60)(6,116,156,121)(7,117,157,122)(8,118,158,123)(9,119,159,124)(10,120,160,125)(11,131,16,126)(12,132,17,127)(13,133,18,128)(14,134,19,129)(15,135,20,130)(26,71,31,66)(27,72,32,67)(28,73,33,68)(29,74,34,69)(30,75,35,70)(36,81,41,76)(37,82,42,77)(38,83,43,78)(39,84,44,79)(40,85,45,80)(46,91,51,86)(47,92,52,87)(48,93,53,88)(49,94,54,89)(50,95,55,90)(96,136,101,141)(97,137,102,142)(98,138,103,143)(99,139,104,144)(100,140,105,145)(106,146,111,151)(107,147,112,152)(108,148,113,153)(109,149,114,154)(110,150,115,155), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,141,16,151)(7,142,17,152)(8,143,18,153)(9,144,19,154)(10,145,20,155)(11,146,156,136)(12,147,157,137)(13,148,158,138)(14,149,159,139)(15,150,160,140)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45)(56,86,66,76)(57,87,67,77)(58,88,68,78)(59,89,69,79)(60,90,70,80)(61,91,71,81)(62,92,72,82)(63,93,73,83)(64,94,74,84)(65,95,75,85)(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), (1,21)(2,22)(3,23)(4,24)(5,25)(6,11)(7,12)(8,13)(9,14)(10,15)(16,156)(17,157)(18,158)(19,159)(20,160)(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)(96,111)(97,112)(98,113)(99,114)(100,115)(101,106)(102,107)(103,108)(104,109)(105,110)(116,131)(117,132)(118,133)(119,134)(120,135)(121,126)(122,127)(123,128)(124,129)(125,130)(136,151)(137,152)(138,153)(139,154)(140,155)(141,146)(142,147)(143,148)(144,149)(145,150), (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), (1,101,21,96)(2,102,22,97)(3,103,23,98)(4,104,24,99)(5,105,25,100)(6,81,156,76)(7,82,157,77)(8,83,158,78)(9,84,159,79)(10,85,160,80)(11,86,16,91)(12,87,17,92)(13,88,18,93)(14,89,19,94)(15,90,20,95)(26,111,31,106)(27,112,32,107)(28,113,33,108)(29,114,34,109)(30,115,35,110)(36,121,41,116)(37,122,42,117)(38,123,43,118)(39,124,44,119)(40,125,45,120)(46,131,51,126)(47,132,52,127)(48,133,53,128)(49,134,54,129)(50,135,55,130)(56,141,61,136)(57,142,62,137)(58,143,63,138)(59,144,64,139)(60,145,65,140)(66,151,71,146)(67,152,72,147)(68,153,73,148)(69,154,74,149)(70,155,75,150), (1,61,21,56)(2,62,22,57)(3,63,23,58)(4,64,24,59)(5,65,25,60)(6,116,156,121)(7,117,157,122)(8,118,158,123)(9,119,159,124)(10,120,160,125)(11,131,16,126)(12,132,17,127)(13,133,18,128)(14,134,19,129)(15,135,20,130)(26,71,31,66)(27,72,32,67)(28,73,33,68)(29,74,34,69)(30,75,35,70)(36,81,41,76)(37,82,42,77)(38,83,43,78)(39,84,44,79)(40,85,45,80)(46,91,51,86)(47,92,52,87)(48,93,53,88)(49,94,54,89)(50,95,55,90)(96,136,101,141)(97,137,102,142)(98,138,103,143)(99,139,104,144)(100,140,105,145)(106,146,111,151)(107,147,112,152)(108,148,113,153)(109,149,114,154)(110,150,115,155), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,141,16,151)(7,142,17,152)(8,143,18,153)(9,144,19,154)(10,145,20,155)(11,146,156,136)(12,147,157,137)(13,148,158,138)(14,149,159,139)(15,150,160,140)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45)(56,86,66,76)(57,87,67,77)(58,88,68,78)(59,89,69,79)(60,90,70,80)(61,91,71,81)(62,92,72,82)(63,93,73,83)(64,94,74,84)(65,95,75,85)(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)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,11),(7,12),(8,13),(9,14),(10,15),(16,156),(17,157),(18,158),(19,159),(20,160),(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),(96,111),(97,112),(98,113),(99,114),(100,115),(101,106),(102,107),(103,108),(104,109),(105,110),(116,131),(117,132),(118,133),(119,134),(120,135),(121,126),(122,127),(123,128),(124,129),(125,130),(136,151),(137,152),(138,153),(139,154),(140,155),(141,146),(142,147),(143,148),(144,149),(145,150)], [(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)], [(1,101,21,96),(2,102,22,97),(3,103,23,98),(4,104,24,99),(5,105,25,100),(6,81,156,76),(7,82,157,77),(8,83,158,78),(9,84,159,79),(10,85,160,80),(11,86,16,91),(12,87,17,92),(13,88,18,93),(14,89,19,94),(15,90,20,95),(26,111,31,106),(27,112,32,107),(28,113,33,108),(29,114,34,109),(30,115,35,110),(36,121,41,116),(37,122,42,117),(38,123,43,118),(39,124,44,119),(40,125,45,120),(46,131,51,126),(47,132,52,127),(48,133,53,128),(49,134,54,129),(50,135,55,130),(56,141,61,136),(57,142,62,137),(58,143,63,138),(59,144,64,139),(60,145,65,140),(66,151,71,146),(67,152,72,147),(68,153,73,148),(69,154,74,149),(70,155,75,150)], [(1,61,21,56),(2,62,22,57),(3,63,23,58),(4,64,24,59),(5,65,25,60),(6,116,156,121),(7,117,157,122),(8,118,158,123),(9,119,159,124),(10,120,160,125),(11,131,16,126),(12,132,17,127),(13,133,18,128),(14,134,19,129),(15,135,20,130),(26,71,31,66),(27,72,32,67),(28,73,33,68),(29,74,34,69),(30,75,35,70),(36,81,41,76),(37,82,42,77),(38,83,43,78),(39,84,44,79),(40,85,45,80),(46,91,51,86),(47,92,52,87),(48,93,53,88),(49,94,54,89),(50,95,55,90),(96,136,101,141),(97,137,102,142),(98,138,103,143),(99,139,104,144),(100,140,105,145),(106,146,111,151),(107,147,112,152),(108,148,113,153),(109,149,114,154),(110,150,115,155)], [(1,46,26,36),(2,47,27,37),(3,48,28,38),(4,49,29,39),(5,50,30,40),(6,141,16,151),(7,142,17,152),(8,143,18,153),(9,144,19,154),(10,145,20,155),(11,146,156,136),(12,147,157,137),(13,148,158,138),(14,149,159,139),(15,150,160,140),(21,51,31,41),(22,52,32,42),(23,53,33,43),(24,54,34,44),(25,55,35,45),(56,86,66,76),(57,87,67,77),(58,88,68,78),(59,89,69,79),(60,90,70,80),(61,91,71,81),(62,92,72,82),(63,93,73,83),(64,94,74,84),(65,95,75,85),(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)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20BD | 20BE | ··· | 20CJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 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 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | Q8 | C4○D4 | C5×Q8 | C5×C4○D4 |
| kernel | C5×C23.37C23 | C2×C4×C20 | C5×C42⋊C2 | Q8×C20 | C5×C22⋊Q8 | C5×C42.C2 | C5×C4⋊Q8 | C23.37C23 | C2×C42 | C42⋊C2 | C4×Q8 | C22⋊Q8 | C42.C2 | C4⋊Q8 | C2×C20 | C20 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 8 | 16 | 16 | 8 | 8 | 4 | 8 | 16 | 32 |
Matrix representation of C5×C23.37C23 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 2 | 40 |
| 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 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 |
| 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,1,0,0,0,0,1,2,0,0,0,40],[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],[0,40,0,0,1,0,0,0,0,0,1,0,0,0,40,40],[32,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,32,0,0,0,0,32] >;
C5×C23.37C23 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{37}C_2^3 % in TeX
G:=Group("C5xC2^3.37C2^3"); // GroupNames label
G:=SmallGroup(320,1535);
// by ID
G=gap.SmallGroup(320,1535);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,3446,856,304]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=1,e^2=f^2=d,g^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,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e,f*g=g*f>;
// generators/relations