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G = C5×C22.47C24order 320 = 26·5

Direct product of C5 and C22.47C24

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

Aliases: C5×C22.47C24, C10.1642+ 1+4, (D4×C20)⋊49C2, (C4×D4)⋊20C10, C4⋊D415C10, C42.C29C10, C422C26C10, C42.47(C2×C10), C42⋊C216C10, C20.324(C4○D4), (C4×C20).288C22, (C2×C10).373C24, (C2×C20).963C23, (D4×C10).221C22, C22.D411C10, C22.47(C23×C10), C23.19(C22×C10), C2.16(C5×2+ 1+4), (C22×C20).458C22, (C22×C10).102C23, (C10×C4⋊C4)⋊49C2, (C2×C4⋊C4)⋊22C10, C4.36(C5×C4○D4), (C5×C4⋊D4)⋊42C2, C4⋊C4.73(C2×C10), C2.26(C10×C4○D4), (C2×D4).34(C2×C10), C10.245(C2×C4○D4), (C5×C42.C2)⋊26C2, C22.11(C5×C4○D4), (C5×C42⋊C2)⋊37C2, (C5×C422C2)⋊17C2, C22⋊C4.23(C2×C10), (C5×C4⋊C4).399C22, (C22×C4).12(C2×C10), (C2×C4).62(C22×C10), (C2×C10).179(C4○D4), (C5×C22.D4)⋊30C2, (C5×C22⋊C4).155C22, SmallGroup(320,1555)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C22.47C24
C1C2C22C2×C10C22×C10C5×C22⋊C4C5×C4⋊D4 — C5×C22.47C24
C1C22 — C5×C22.47C24
C1C2×C10 — C5×C22.47C24

Generators and relations for C5×C22.47C24
 G = < a,b,c,d,e,f,g | a5=b2=c2=d2=f2=1, e2=cb=bc, 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=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 362 in 238 conjugacy classes, 150 normal (62 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×11], C5, C2×C4 [×5], C2×C4 [×6], C2×C4 [×8], D4 [×10], C23 [×2], C23 [×2], C10 [×3], C10 [×5], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C20 [×2], C20 [×10], C2×C10, C2×C10 [×2], C2×C10 [×11], C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22.D4 [×2], C42.C2, C422C2 [×2], C2×C20 [×5], C2×C20 [×6], C2×C20 [×8], C5×D4 [×10], C22×C10 [×2], C22×C10 [×2], C22.47C24, C4×C20, C4×C20 [×2], C5×C22⋊C4 [×2], C5×C22⋊C4 [×8], C5×C4⋊C4 [×4], C5×C4⋊C4 [×6], C22×C20 [×2], C22×C20 [×4], D4×C10 [×2], D4×C10 [×4], C10×C4⋊C4, C5×C42⋊C2, D4×C20 [×2], D4×C20 [×2], C5×C4⋊D4 [×2], C5×C4⋊D4 [×2], C5×C22.D4 [×2], C5×C42.C2, C5×C422C2 [×2], C5×C22.47C24
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], C22.47C24, C5×C4○D4 [×4], C23×C10, C10×C4○D4 [×2], C5×2+ 1+4, C5×C22.47C24

Smallest permutation representation of C5×C22.47C24
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 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)
(1 136)(2 137)(3 138)(4 139)(5 140)(6 40)(7 36)(8 37)(9 38)(10 39)(11 48)(12 49)(13 50)(14 46)(15 47)(16 41)(17 42)(18 43)(19 44)(20 45)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(51 157)(52 158)(53 159)(54 160)(55 156)(56 101)(57 102)(58 103)(59 104)(60 105)(61 96)(62 97)(63 98)(64 99)(65 100)(66 114)(67 115)(68 111)(69 112)(70 113)(71 108)(72 109)(73 110)(74 106)(75 107)(76 127)(77 128)(78 129)(79 130)(80 126)(81 135)(82 131)(83 132)(84 133)(85 134)(86 120)(87 116)(88 117)(89 118)(90 119)(91 122)(92 123)(93 124)(94 125)(95 121)
(1 107 27 96)(2 108 28 97)(3 109 29 98)(4 110 30 99)(5 106 26 100)(6 94 14 85)(7 95 15 81)(8 91 11 82)(9 92 12 83)(10 93 13 84)(16 87 156 76)(17 88 157 77)(18 89 158 78)(19 90 159 79)(20 86 160 80)(21 115 35 101)(22 111 31 102)(23 112 32 103)(24 113 33 104)(25 114 34 105)(36 127 47 116)(37 128 48 117)(38 129 49 118)(39 130 50 119)(40 126 46 120)(41 135 55 121)(42 131 51 122)(43 132 52 123)(44 133 53 124)(45 134 54 125)(56 147 67 136)(57 148 68 137)(58 149 69 138)(59 150 70 139)(60 146 66 140)(61 155 75 141)(62 151 71 142)(63 152 72 143)(64 153 73 144)(65 154 74 145)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 134)(7 135)(8 131)(9 132)(10 133)(11 122)(12 123)(13 124)(14 125)(15 121)(16 127)(17 128)(18 129)(19 130)(20 126)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(96 136)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 140 160 154)(7 136 156 155)(8 137 157 151)(9 138 158 152)(10 139 159 153)(11 148 17 142)(12 149 18 143)(13 150 19 144)(14 146 20 145)(15 147 16 141)(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)(96 116 115 135)(97 117 111 131)(98 118 112 132)(99 119 113 133)(100 120 114 134)(101 121 107 127)(102 122 108 128)(103 123 109 129)(104 124 110 130)(105 125 106 126)

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

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

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

125 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4J4K···4P5A5B5C5D10A···10L10M···10T10U···10AF20A···20AN20AO···20BL
order1222222224···44···4555510···1010···1010···1020···2020···20
size1111224442···24···411111···12···24···42···24···4

125 irreducible representations

dim1111111111111111222244
type+++++++++
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10C4○D4C4○D4C5×C4○D4C5×C4○D42+ 1+4C5×2+ 1+4
kernelC5×C22.47C24C10×C4⋊C4C5×C42⋊C2D4×C20C5×C4⋊D4C5×C22.D4C5×C42.C2C5×C422C2C22.47C24C2×C4⋊C4C42⋊C2C4×D4C4⋊D4C22.D4C42.C2C422C2C20C2×C10C4C22C10C2
# reps11144212444161684844161614

Matrix representation of C5×C22.47C24 in GL4(𝔽41) generated by

1000
0100
00100
00010
,
1000
0100
00400
00040
,
40000
04000
0010
0001
,
03200
9000
003218
00329
,
32000
03200
003218
0009
,
0100
1000
0010
0001
,
40000
04000
00139
00140
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[0,9,0,0,32,0,0,0,0,0,32,32,0,0,18,9],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,18,9],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,1,0,0,39,40] >;

C5×C22.47C24 in GAP, Magma, Sage, TeX

C_5\times C_2^2._{47}C_2^4
% in TeX

G:=Group("C5xC2^2.47C2^4");
// GroupNames label

G:=SmallGroup(320,1555);
// by ID

G=gap.SmallGroup(320,1555);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1688,3446,1242,304]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=f^2=1,e^2=c*b=b*c,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=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