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G = C5×C22.50C24order 320 = 26·5

Direct product of C5 and C22.50C24

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

Aliases: C5×C22.50C24, C10.1212- 1+4, C4⋊Q817C10, (Q8×C20)⋊36C2, (C4×Q8)⋊16C10, (D4×C20).28C2, (C4×D4).13C10, C22⋊Q818C10, C422C27C10, C42.50(C2×C10), C4.4D4.8C10, C42⋊C218C10, 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×C422C2)⋊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

C1C22 — C5×C22.50C24
C1C2C22C2×C10C22×C10C5×C22⋊C4C5×C422C2 — C5×C22.50C24
C1C22 — C5×C22.50C24
C1C2×C10 — C5×C22.50C24

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 [×3], C2 [×2], C4 [×4], C4 [×11], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×10], C2×C4 [×4], D4 [×2], Q8 [×6], C23 [×2], C10 [×3], C10 [×2], C42, C42 [×6], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×10], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], C20 [×4], C20 [×11], C2×C10, C2×C10 [×6], C42⋊C2 [×2], C4×D4, C4×Q8, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C422C2 [×4], C4⋊Q8, C2×C20 [×3], C2×C20 [×10], C2×C20 [×4], C5×D4 [×2], C5×Q8 [×6], C22×C10 [×2], C22.50C24, C4×C20, C4×C20 [×6], C5×C22⋊C4 [×10], C5×C4⋊C4 [×2], C5×C4⋊C4 [×10], C22×C20 [×2], D4×C10, Q8×C10, Q8×C10 [×2], C5×C42⋊C2 [×2], D4×C20, Q8×C20, Q8×C20 [×2], C5×C22⋊Q8 [×2], C5×C4.4D4 [×2], C5×C422C2 [×4], C5×C4⋊Q8, C5×C22.50C24
Quotients: C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C4○D4 [×4], C24, C2×C10 [×35], C2×C4○D4 [×2], 2- 1+4, C22×C10 [×15], C22.50C24, C5×C4○D4 [×4], C23×C10, C10×C4○D4 [×2], C5×2- 1+4, C5×C22.50C24

Smallest permutation representation of 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 2A2B2C2D2E4A···4L4M···4S5A5B5C5D10A···10L10M···10T20A···20AV20AW···20BX
order1222224···44···4555510···1010···1020···2020···20
size1111442···24···411111···14···42···24···4

125 irreducible representations

dim11111111111111112244
type++++++++-
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10C4○D4C5×C4○D42- 1+4C5×2- 1+4
kernelC5×C22.50C24C5×C42⋊C2D4×C20Q8×C20C5×C22⋊Q8C5×C4.4D4C5×C422C2C5×C4⋊Q8C22.50C24C42⋊C2C4×D4C4×Q8C22⋊Q8C4.4D4C422C2C4⋊Q8C20C4C10C2
# reps12132241484128816483214

Matrix representation of C5×C22.50C24 in GL4(𝔽41) generated by

1000
0100
00180
00018
,
40000
04000
0010
0001
,
1000
0100
00400
00040
,
40000
0100
0001
0010
,
0100
40000
00320
00032
,
9000
0900
0010
00040
,
32000
0900
0010
0001
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

׿
×
𝔽