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, C4.10(Q8×C10), C20.99(C2×Q8), 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), (C4×C20).376C22, (C2×C20).662C23, (C2×C10).353C24, C42⋊C2.11C10, C23.35(C22×C10), C22.27(C23×C10), (Q8×C10).269C22, (C22×C20).598C22, (C22×C10).260C23, (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), (C22×C4).125(C2×C10), (C5×C42⋊C2).25C2, (C5×C22⋊C4).148C22, SmallGroup(320,1535)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 274 in 222 conjugacy classes, 170 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×16], C2×C4 [×4], Q8 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×2], C42 [×6], C22⋊C4 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×4], C20 [×8], C20 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C2×C20 [×2], C2×C20 [×16], C2×C20 [×4], C5×Q8 [×8], C22×C10, C23.37C23, C4×C20 [×2], C4×C20 [×6], C5×C22⋊C4 [×4], C5×C4⋊C4 [×16], C22×C20, C22×C20 [×2], Q8×C10 [×4], C2×C4×C20, C5×C42⋊C2 [×2], Q8×C20 [×4], C5×C22⋊Q8 [×4], C5×C42.C2 [×2], C5×C4⋊Q8 [×2], C5×C23.37C23
Quotients:
C1, C2 [×15], C22 [×35], C5, Q8 [×4], C23 [×15], C10 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C2×C10 [×35], C22×Q8, C2×C4○D4 [×2], C5×Q8 [×4], C22×C10 [×15], C23.37C23, Q8×C10 [×6], C5×C4○D4 [×4], C23×C10, Q8×C2×C10, C10×C4○D4 [×2], C5×C23.37C23
Generators and relations
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 >
►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 14)(7 15)(8 11)(9 12)(10 13)(16 156)(17 157)(18 158)(19 159)(20 160)(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)(96 107)(97 108)(98 109)(99 110)(100 106)(101 115)(102 111)(103 112)(104 113)(105 114)(116 127)(117 128)(118 129)(119 130)(120 126)(121 135)(122 131)(123 132)(124 133)(125 134)(136 147)(137 148)(138 149)(139 150)(140 146)(141 155)(142 151)(143 152)(144 153)(145 154)
(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)
(1 101 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 61 21 56)(2 62 22 57)(3 63 23 58)(4 64 24 59)(5 65 25 60)(6 126 20 134)(7 127 16 135)(8 128 17 131)(9 129 18 132)(10 130 19 133)(11 117 157 122)(12 118 158 123)(13 119 159 124)(14 120 160 125)(15 116 156 121)(26 74 34 66)(27 75 35 67)(28 71 31 68)(29 72 32 69)(30 73 33 70)(36 81 41 76)(37 82 42 77)(38 83 43 78)(39 84 44 79)(40 85 45 80)(46 94 54 86)(47 95 55 87)(48 91 51 88)(49 92 52 89)(50 93 53 90)(96 136 101 141)(97 137 102 142)(98 138 103 143)(99 139 104 144)(100 140 105 145)(106 146 114 154)(107 147 115 155)(108 148 111 151)(109 149 112 152)(110 150 113 153)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 154 160 140)(7 155 156 136)(8 151 157 137)(9 152 158 138)(10 153 159 139)(11 142 17 148)(12 143 18 149)(13 144 19 150)(14 145 20 146)(15 141 16 147)(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)(96 135 115 116)(97 131 111 117)(98 132 112 118)(99 133 113 119)(100 134 114 120)(101 127 107 121)(102 128 108 122)(103 129 109 123)(104 130 110 124)(105 126 106 125)
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)(6,14)(7,15)(8,11)(9,12)(10,13)(16,156)(17,157)(18,158)(19,159)(20,160)(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)(96,107)(97,108)(98,109)(99,110)(100,106)(101,115)(102,111)(103,112)(104,113)(105,114)(116,127)(117,128)(118,129)(119,130)(120,126)(121,135)(122,131)(123,132)(124,133)(125,134)(136,147)(137,148)(138,149)(139,150)(140,146)(141,155)(142,151)(143,152)(144,153)(145,154), (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), (1,101,21,96)(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), (1,61,21,56)(2,62,22,57)(3,63,23,58)(4,64,24,59)(5,65,25,60)(6,126,20,134)(7,127,16,135)(8,128,17,131)(9,129,18,132)(10,130,19,133)(11,117,157,122)(12,118,158,123)(13,119,159,124)(14,120,160,125)(15,116,156,121)(26,74,34,66)(27,75,35,67)(28,71,31,68)(29,72,32,69)(30,73,33,70)(36,81,41,76)(37,82,42,77)(38,83,43,78)(39,84,44,79)(40,85,45,80)(46,94,54,86)(47,95,55,87)(48,91,51,88)(49,92,52,89)(50,93,53,90)(96,136,101,141)(97,137,102,142)(98,138,103,143)(99,139,104,144)(100,140,105,145)(106,146,114,154)(107,147,115,155)(108,148,111,151)(109,149,112,152)(110,150,113,153), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,154,160,140)(7,155,156,136)(8,151,157,137)(9,152,158,138)(10,153,159,139)(11,142,17,148)(12,143,18,149)(13,144,19,150)(14,145,20,146)(15,141,16,147)(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)(96,135,115,116)(97,131,111,117)(98,132,112,118)(99,133,113,119)(100,134,114,120)(101,127,107,121)(102,128,108,122)(103,129,109,123)(104,130,110,124)(105,126,106,125)>;
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)(6,14)(7,15)(8,11)(9,12)(10,13)(16,156)(17,157)(18,158)(19,159)(20,160)(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)(96,107)(97,108)(98,109)(99,110)(100,106)(101,115)(102,111)(103,112)(104,113)(105,114)(116,127)(117,128)(118,129)(119,130)(120,126)(121,135)(122,131)(123,132)(124,133)(125,134)(136,147)(137,148)(138,149)(139,150)(140,146)(141,155)(142,151)(143,152)(144,153)(145,154), (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), (1,101,21,96)(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), (1,61,21,56)(2,62,22,57)(3,63,23,58)(4,64,24,59)(5,65,25,60)(6,126,20,134)(7,127,16,135)(8,128,17,131)(9,129,18,132)(10,130,19,133)(11,117,157,122)(12,118,158,123)(13,119,159,124)(14,120,160,125)(15,116,156,121)(26,74,34,66)(27,75,35,67)(28,71,31,68)(29,72,32,69)(30,73,33,70)(36,81,41,76)(37,82,42,77)(38,83,43,78)(39,84,44,79)(40,85,45,80)(46,94,54,86)(47,95,55,87)(48,91,51,88)(49,92,52,89)(50,93,53,90)(96,136,101,141)(97,137,102,142)(98,138,103,143)(99,139,104,144)(100,140,105,145)(106,146,114,154)(107,147,115,155)(108,148,111,151)(109,149,112,152)(110,150,113,153), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,154,160,140)(7,155,156,136)(8,151,157,137)(9,152,158,138)(10,153,159,139)(11,142,17,148)(12,143,18,149)(13,144,19,150)(14,145,20,146)(15,141,16,147)(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)(96,135,115,116)(97,131,111,117)(98,132,112,118)(99,133,113,119)(100,134,114,120)(101,127,107,121)(102,128,108,122)(103,129,109,123)(104,130,110,124)(105,126,106,125) );
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),(6,14),(7,15),(8,11),(9,12),(10,13),(16,156),(17,157),(18,158),(19,159),(20,160),(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),(96,107),(97,108),(98,109),(99,110),(100,106),(101,115),(102,111),(103,112),(104,113),(105,114),(116,127),(117,128),(118,129),(119,130),(120,126),(121,135),(122,131),(123,132),(124,133),(125,134),(136,147),(137,148),(138,149),(139,150),(140,146),(141,155),(142,151),(143,152),(144,153),(145,154)], [(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)], [(1,101,21,96),(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)], [(1,61,21,56),(2,62,22,57),(3,63,23,58),(4,64,24,59),(5,65,25,60),(6,126,20,134),(7,127,16,135),(8,128,17,131),(9,129,18,132),(10,130,19,133),(11,117,157,122),(12,118,158,123),(13,119,159,124),(14,120,160,125),(15,116,156,121),(26,74,34,66),(27,75,35,67),(28,71,31,68),(29,72,32,69),(30,73,33,70),(36,81,41,76),(37,82,42,77),(38,83,43,78),(39,84,44,79),(40,85,45,80),(46,94,54,86),(47,95,55,87),(48,91,51,88),(49,92,52,89),(50,93,53,90),(96,136,101,141),(97,137,102,142),(98,138,103,143),(99,139,104,144),(100,140,105,145),(106,146,114,154),(107,147,115,155),(108,148,111,151),(109,149,112,152),(110,150,113,153)], [(1,55,35,36),(2,51,31,37),(3,52,32,38),(4,53,33,39),(5,54,34,40),(6,154,160,140),(7,155,156,136),(8,151,157,137),(9,152,158,138),(10,153,159,139),(11,142,17,148),(12,143,18,149),(13,144,19,150),(14,145,20,146),(15,141,16,147),(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),(96,135,115,116),(97,131,111,117),(98,132,112,118),(99,133,113,119),(100,134,114,120),(101,127,107,121),(102,128,108,122),(103,129,109,123),(104,130,110,124),(105,126,106,125)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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