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G = C5×C23.37C23order 320 = 26·5

Direct product of C5 and C23.37C23

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C23.37C23, C4⋊Q820C10, (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.C214C10, 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

C1C22 — C5×C23.37C23
C1C2C22C2×C10C2×C20C5×C22⋊C4C5×C22⋊Q8 — C5×C23.37C23
C1C22 — C5×C23.37C23
C1C2×C20 — C5×C23.37C23

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 [×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

Smallest permutation representation of 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 30)(2 26)(3 27)(4 28)(5 29)(6 11)(7 12)(8 13)(9 14)(10 15)(16 156)(17 157)(18 158)(19 159)(20 160)(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)(96 103)(97 104)(98 105)(99 101)(100 102)(106 111)(107 112)(108 113)(109 114)(110 115)(116 123)(117 124)(118 125)(119 121)(120 122)(126 131)(127 132)(128 133)(129 134)(130 135)(136 143)(137 144)(138 145)(139 141)(140 142)(146 151)(147 152)(148 153)(149 154)(150 155)
(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)
(1 110 30 96)(2 106 26 97)(3 107 27 98)(4 108 28 99)(5 109 29 100)(6 90 156 76)(7 86 157 77)(8 87 158 78)(9 88 159 79)(10 89 160 80)(11 95 16 83)(12 91 17 84)(13 92 18 85)(14 93 19 81)(15 94 20 82)(21 113 33 101)(22 114 34 102)(23 115 35 103)(24 111 31 104)(25 112 32 105)(36 130 50 116)(37 126 46 117)(38 127 47 118)(39 128 48 119)(40 129 49 120)(41 133 53 121)(42 134 54 122)(43 135 55 123)(44 131 51 124)(45 132 52 125)(56 150 70 136)(57 146 66 137)(58 147 67 138)(59 148 68 139)(60 149 69 140)(61 153 73 141)(62 154 74 142)(63 155 75 143)(64 151 71 144)(65 152 72 145)
(1 70 30 56)(2 66 26 57)(3 67 27 58)(4 68 28 59)(5 69 29 60)(6 116 156 130)(7 117 157 126)(8 118 158 127)(9 119 159 128)(10 120 160 129)(11 123 16 135)(12 124 17 131)(13 125 18 132)(14 121 19 133)(15 122 20 134)(21 73 33 61)(22 74 34 62)(23 75 35 63)(24 71 31 64)(25 72 32 65)(36 90 50 76)(37 86 46 77)(38 87 47 78)(39 88 48 79)(40 89 49 80)(41 93 53 81)(42 94 54 82)(43 95 55 83)(44 91 51 84)(45 92 52 85)(96 136 110 150)(97 137 106 146)(98 138 107 147)(99 139 108 148)(100 140 109 149)(101 141 113 153)(102 142 114 154)(103 143 115 155)(104 144 111 151)(105 145 112 152)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 150 16 143)(7 146 17 144)(8 147 18 145)(9 148 19 141)(10 149 20 142)(11 155 156 136)(12 151 157 137)(13 152 158 138)(14 153 159 139)(15 154 160 140)(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)(96 135 115 116)(97 131 111 117)(98 132 112 118)(99 133 113 119)(100 134 114 120)(101 128 108 121)(102 129 109 122)(103 130 110 123)(104 126 106 124)(105 127 107 125)

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), (1,30)(2,26)(3,27)(4,28)(5,29)(6,11)(7,12)(8,13)(9,14)(10,15)(16,156)(17,157)(18,158)(19,159)(20,160)(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)(96,103)(97,104)(98,105)(99,101)(100,102)(106,111)(107,112)(108,113)(109,114)(110,115)(116,123)(117,124)(118,125)(119,121)(120,122)(126,131)(127,132)(128,133)(129,134)(130,135)(136,143)(137,144)(138,145)(139,141)(140,142)(146,151)(147,152)(148,153)(149,154)(150,155), (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), (1,110,30,96)(2,106,26,97)(3,107,27,98)(4,108,28,99)(5,109,29,100)(6,90,156,76)(7,86,157,77)(8,87,158,78)(9,88,159,79)(10,89,160,80)(11,95,16,83)(12,91,17,84)(13,92,18,85)(14,93,19,81)(15,94,20,82)(21,113,33,101)(22,114,34,102)(23,115,35,103)(24,111,31,104)(25,112,32,105)(36,130,50,116)(37,126,46,117)(38,127,47,118)(39,128,48,119)(40,129,49,120)(41,133,53,121)(42,134,54,122)(43,135,55,123)(44,131,51,124)(45,132,52,125)(56,150,70,136)(57,146,66,137)(58,147,67,138)(59,148,68,139)(60,149,69,140)(61,153,73,141)(62,154,74,142)(63,155,75,143)(64,151,71,144)(65,152,72,145), (1,70,30,56)(2,66,26,57)(3,67,27,58)(4,68,28,59)(5,69,29,60)(6,116,156,130)(7,117,157,126)(8,118,158,127)(9,119,159,128)(10,120,160,129)(11,123,16,135)(12,124,17,131)(13,125,18,132)(14,121,19,133)(15,122,20,134)(21,73,33,61)(22,74,34,62)(23,75,35,63)(24,71,31,64)(25,72,32,65)(36,90,50,76)(37,86,46,77)(38,87,47,78)(39,88,48,79)(40,89,49,80)(41,93,53,81)(42,94,54,82)(43,95,55,83)(44,91,51,84)(45,92,52,85)(96,136,110,150)(97,137,106,146)(98,138,107,147)(99,139,108,148)(100,140,109,149)(101,141,113,153)(102,142,114,154)(103,143,115,155)(104,144,111,151)(105,145,112,152), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,150,16,143)(7,146,17,144)(8,147,18,145)(9,148,19,141)(10,149,20,142)(11,155,156,136)(12,151,157,137)(13,152,158,138)(14,153,159,139)(15,154,160,140)(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)(96,135,115,116)(97,131,111,117)(98,132,112,118)(99,133,113,119)(100,134,114,120)(101,128,108,121)(102,129,109,122)(103,130,110,123)(104,126,106,124)(105,127,107,125)>;

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), (1,30)(2,26)(3,27)(4,28)(5,29)(6,11)(7,12)(8,13)(9,14)(10,15)(16,156)(17,157)(18,158)(19,159)(20,160)(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)(96,103)(97,104)(98,105)(99,101)(100,102)(106,111)(107,112)(108,113)(109,114)(110,115)(116,123)(117,124)(118,125)(119,121)(120,122)(126,131)(127,132)(128,133)(129,134)(130,135)(136,143)(137,144)(138,145)(139,141)(140,142)(146,151)(147,152)(148,153)(149,154)(150,155), (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), (1,110,30,96)(2,106,26,97)(3,107,27,98)(4,108,28,99)(5,109,29,100)(6,90,156,76)(7,86,157,77)(8,87,158,78)(9,88,159,79)(10,89,160,80)(11,95,16,83)(12,91,17,84)(13,92,18,85)(14,93,19,81)(15,94,20,82)(21,113,33,101)(22,114,34,102)(23,115,35,103)(24,111,31,104)(25,112,32,105)(36,130,50,116)(37,126,46,117)(38,127,47,118)(39,128,48,119)(40,129,49,120)(41,133,53,121)(42,134,54,122)(43,135,55,123)(44,131,51,124)(45,132,52,125)(56,150,70,136)(57,146,66,137)(58,147,67,138)(59,148,68,139)(60,149,69,140)(61,153,73,141)(62,154,74,142)(63,155,75,143)(64,151,71,144)(65,152,72,145), (1,70,30,56)(2,66,26,57)(3,67,27,58)(4,68,28,59)(5,69,29,60)(6,116,156,130)(7,117,157,126)(8,118,158,127)(9,119,159,128)(10,120,160,129)(11,123,16,135)(12,124,17,131)(13,125,18,132)(14,121,19,133)(15,122,20,134)(21,73,33,61)(22,74,34,62)(23,75,35,63)(24,71,31,64)(25,72,32,65)(36,90,50,76)(37,86,46,77)(38,87,47,78)(39,88,48,79)(40,89,49,80)(41,93,53,81)(42,94,54,82)(43,95,55,83)(44,91,51,84)(45,92,52,85)(96,136,110,150)(97,137,106,146)(98,138,107,147)(99,139,108,148)(100,140,109,149)(101,141,113,153)(102,142,114,154)(103,143,115,155)(104,144,111,151)(105,145,112,152), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,150,16,143)(7,146,17,144)(8,147,18,145)(9,148,19,141)(10,149,20,142)(11,155,156,136)(12,151,157,137)(13,152,158,138)(14,153,159,139)(15,154,160,140)(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)(96,135,115,116)(97,131,111,117)(98,132,112,118)(99,133,113,119)(100,134,114,120)(101,128,108,121)(102,129,109,122)(103,130,110,123)(104,126,106,124)(105,127,107,125) );

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)], [(1,30),(2,26),(3,27),(4,28),(5,29),(6,11),(7,12),(8,13),(9,14),(10,15),(16,156),(17,157),(18,158),(19,159),(20,160),(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),(96,103),(97,104),(98,105),(99,101),(100,102),(106,111),(107,112),(108,113),(109,114),(110,115),(116,123),(117,124),(118,125),(119,121),(120,122),(126,131),(127,132),(128,133),(129,134),(130,135),(136,143),(137,144),(138,145),(139,141),(140,142),(146,151),(147,152),(148,153),(149,154),(150,155)], [(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)], [(1,110,30,96),(2,106,26,97),(3,107,27,98),(4,108,28,99),(5,109,29,100),(6,90,156,76),(7,86,157,77),(8,87,158,78),(9,88,159,79),(10,89,160,80),(11,95,16,83),(12,91,17,84),(13,92,18,85),(14,93,19,81),(15,94,20,82),(21,113,33,101),(22,114,34,102),(23,115,35,103),(24,111,31,104),(25,112,32,105),(36,130,50,116),(37,126,46,117),(38,127,47,118),(39,128,48,119),(40,129,49,120),(41,133,53,121),(42,134,54,122),(43,135,55,123),(44,131,51,124),(45,132,52,125),(56,150,70,136),(57,146,66,137),(58,147,67,138),(59,148,68,139),(60,149,69,140),(61,153,73,141),(62,154,74,142),(63,155,75,143),(64,151,71,144),(65,152,72,145)], [(1,70,30,56),(2,66,26,57),(3,67,27,58),(4,68,28,59),(5,69,29,60),(6,116,156,130),(7,117,157,126),(8,118,158,127),(9,119,159,128),(10,120,160,129),(11,123,16,135),(12,124,17,131),(13,125,18,132),(14,121,19,133),(15,122,20,134),(21,73,33,61),(22,74,34,62),(23,75,35,63),(24,71,31,64),(25,72,32,65),(36,90,50,76),(37,86,46,77),(38,87,47,78),(39,88,48,79),(40,89,49,80),(41,93,53,81),(42,94,54,82),(43,95,55,83),(44,91,51,84),(45,92,52,85),(96,136,110,150),(97,137,106,146),(98,138,107,147),(99,139,108,148),(100,140,109,149),(101,141,113,153),(102,142,114,154),(103,143,115,155),(104,144,111,151),(105,145,112,152)], [(1,55,35,36),(2,51,31,37),(3,52,32,38),(4,53,33,39),(5,54,34,40),(6,150,16,143),(7,146,17,144),(8,147,18,145),(9,148,19,141),(10,149,20,142),(11,155,156,136),(12,151,157,137),(13,152,158,138),(14,153,159,139),(15,154,160,140),(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),(96,135,115,116),(97,131,111,117),(98,132,112,118),(99,133,113,119),(100,134,114,120),(101,128,108,121),(102,129,109,122),(103,130,110,123),(104,126,106,124),(105,127,107,125)])

140 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N4O···4V5A5B5C5D10A···10L10M···10T20A···20P20Q···20BD20BE···20CJ
order12222244444···44···4555510···1010···1020···2020···2020···20
size11112211112···24···411111···12···21···12···24···4

140 irreducible representations

dim111111111111112222
type+++++++-
imageC1C2C2C2C2C2C2C5C10C10C10C10C10C10Q8C4○D4C5×Q8C5×C4○D4
kernelC5×C23.37C23C2×C4×C20C5×C42⋊C2Q8×C20C5×C22⋊Q8C5×C42.C2C5×C4⋊Q8C23.37C23C2×C42C42⋊C2C4×Q8C22⋊Q8C42.C2C4⋊Q8C2×C20C20C2×C4C4
# reps1124422448161688481632

Matrix representation of C5×C23.37C23 in GL4(𝔽41) generated by

10000
01000
00370
00037
,
40000
0100
0010
00240
,
40000
04000
00400
00040
,
40000
04000
0010
0001
,
0100
40000
00140
00040
,
32000
0900
0010
0001
,
9000
0900
00320
00032
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

׿
×
𝔽