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G = C5×Dic35D4order 480 = 25·3·5

Direct product of C5 and Dic35D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×Dic35D4, D125C20, C33(D4×C20), C41(S3×C20), C1539(C4×D4), C2013(C4×S3), C122(C2×C20), C6030(C2×C4), D63(C2×C20), D6⋊C412C10, (C5×D12)⋊17C4, Dic35(C5×D4), C6.24(D4×C10), (C5×Dic3)⋊20D4, (C4×Dic3)⋊3C10, (C2×D12).7C10, C30.360(C2×D4), C10.177(S3×D4), (C2×C20).277D6, (Dic3×C20)⋊15C2, (C10×D12).17C2, C6.11(C22×C20), C30.268(C4○D4), (C2×C30).413C23, C30.202(C22×C4), (C2×C60).352C22, C10.49(Q83S3), (C10×Dic3).220C22, C4⋊C48(C5×S3), C2.4(C5×S3×D4), (C3×C4⋊C4)⋊4C10, (C5×C4⋊C4)⋊17S3, (S3×C2×C4)⋊12C10, (S3×C2×C20)⋊28C2, C2.13(S3×C2×C20), (C15×C4⋊C4)⋊22C2, (C5×D6⋊C4)⋊34C2, C10.138(S3×C2×C4), C6.32(C5×C4○D4), (S3×C10)⋊25(C2×C4), (C2×C4).44(S3×C10), C22.18(S3×C2×C10), C2.2(C5×Q83S3), (C2×C12).23(C2×C10), (S3×C2×C10).111C22, (C2×C6).34(C22×C10), (C22×S3).20(C2×C10), (C2×C10).347(C22×S3), (C2×Dic3).49(C2×C10), SmallGroup(480,772)

Series: Derived Chief Lower central Upper central

C1C6 — C5×Dic35D4
C1C3C6C2×C6C2×C30S3×C2×C10C10×D12 — C5×Dic35D4
C3C6 — C5×Dic35D4
C1C2×C10C5×C4⋊C4

Generators and relations for C5×Dic35D4
 G = < a,b,c,d,e | a5=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 436 in 188 conjugacy classes, 86 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×8], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×4], C23 [×2], C10 [×3], C10 [×4], Dic3 [×2], Dic3, C12 [×2], C12 [×2], D6 [×4], D6 [×4], C2×C6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C20 [×2], C20 [×5], C2×C10, C2×C10 [×8], C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3 [×2], C5×S3 [×4], C30 [×3], C4×D4, C2×C20, C2×C20 [×2], C2×C20 [×6], C5×D4 [×4], C22×C10 [×2], C4×Dic3, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4 [×2], C2×D12, C5×Dic3 [×2], C5×Dic3, C60 [×2], C60 [×2], S3×C10 [×4], S3×C10 [×4], C2×C30, C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20 [×2], D4×C10, Dic35D4, S3×C20 [×4], C5×D12 [×4], C10×Dic3 [×2], C2×C60, C2×C60 [×2], S3×C2×C10 [×2], D4×C20, Dic3×C20, C5×D6⋊C4 [×2], C15×C4⋊C4, S3×C2×C20 [×2], C10×D12, C5×Dic35D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], D4 [×2], C23, C10 [×7], D6 [×3], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×S3 [×2], C22×S3, C5×S3, C4×D4, C2×C20 [×6], C5×D4 [×2], C22×C10, S3×C2×C4, S3×D4, Q83S3, S3×C10 [×3], C22×C20, D4×C10, C5×C4○D4, Dic35D4, S3×C20 [×2], S3×C2×C10, D4×C20, S3×C2×C20, C5×S3×D4, C5×Q83S3, C5×Dic35D4

Smallest permutation representation of C5×Dic35D4
On 240 points
Generators in S240
(1 60 48 36 24)(2 55 43 31 19)(3 56 44 32 20)(4 57 45 33 21)(5 58 46 34 22)(6 59 47 35 23)(7 231 219 207 195)(8 232 220 208 196)(9 233 221 209 197)(10 234 222 210 198)(11 229 217 205 193)(12 230 218 206 194)(13 61 49 37 25)(14 62 50 38 26)(15 63 51 39 27)(16 64 52 40 28)(17 65 53 41 29)(18 66 54 42 30)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)(73 121 109 97 85)(74 122 110 98 86)(75 123 111 99 87)(76 124 112 100 88)(77 125 113 101 89)(78 126 114 102 90)(127 175 163 151 139)(128 176 164 152 140)(129 177 165 153 141)(130 178 166 154 142)(131 179 167 155 143)(132 180 168 156 144)(133 181 169 157 145)(134 182 170 158 146)(135 183 171 159 147)(136 184 172 160 148)(137 185 173 161 149)(138 186 174 162 150)(187 235 223 211 199)(188 236 224 212 200)(189 237 225 213 201)(190 238 226 214 202)(191 239 227 215 203)(192 240 228 216 204)
(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)(157 158 159 160 161 162)(163 164 165 166 167 168)(169 170 171 172 173 174)(175 176 177 178 179 180)(181 182 183 184 185 186)(187 188 189 190 191 192)(193 194 195 196 197 198)(199 200 201 202 203 204)(205 206 207 208 209 210)(211 212 213 214 215 216)(217 218 219 220 221 222)(223 224 225 226 227 228)(229 230 231 232 233 234)(235 236 237 238 239 240)
(1 69 4 72)(2 68 5 71)(3 67 6 70)(7 181 10 184)(8 186 11 183)(9 185 12 182)(13 78 16 75)(14 77 17 74)(15 76 18 73)(19 80 22 83)(20 79 23 82)(21 84 24 81)(25 90 28 87)(26 89 29 86)(27 88 30 85)(31 92 34 95)(32 91 35 94)(33 96 36 93)(37 102 40 99)(38 101 41 98)(39 100 42 97)(43 104 46 107)(44 103 47 106)(45 108 48 105)(49 114 52 111)(50 113 53 110)(51 112 54 109)(55 116 58 119)(56 115 59 118)(57 120 60 117)(61 126 64 123)(62 125 65 122)(63 124 66 121)(127 188 130 191)(128 187 131 190)(129 192 132 189)(133 198 136 195)(134 197 137 194)(135 196 138 193)(139 200 142 203)(140 199 143 202)(141 204 144 201)(145 210 148 207)(146 209 149 206)(147 208 150 205)(151 212 154 215)(152 211 155 214)(153 216 156 213)(157 222 160 219)(158 221 161 218)(159 220 162 217)(163 224 166 227)(164 223 167 226)(165 228 168 225)(169 234 172 231)(170 233 173 230)(171 232 174 229)(175 236 178 239)(176 235 179 238)(177 240 180 237)
(1 137 17 132)(2 138 18 127)(3 133 13 128)(4 134 14 129)(5 135 15 130)(6 136 16 131)(7 123 238 118)(8 124 239 119)(9 125 240 120)(10 126 235 115)(11 121 236 116)(12 122 237 117)(19 150 30 139)(20 145 25 140)(21 146 26 141)(22 147 27 142)(23 148 28 143)(24 149 29 144)(31 162 42 151)(32 157 37 152)(33 158 38 153)(34 159 39 154)(35 160 40 155)(36 161 41 156)(43 174 54 163)(44 169 49 164)(45 170 50 165)(46 171 51 166)(47 172 52 167)(48 173 53 168)(55 186 66 175)(56 181 61 176)(57 182 62 177)(58 183 63 178)(59 184 64 179)(60 185 65 180)(67 198 78 187)(68 193 73 188)(69 194 74 189)(70 195 75 190)(71 196 76 191)(72 197 77 192)(79 210 90 199)(80 205 85 200)(81 206 86 201)(82 207 87 202)(83 208 88 203)(84 209 89 204)(91 222 102 211)(92 217 97 212)(93 218 98 213)(94 219 99 214)(95 220 100 215)(96 221 101 216)(103 234 114 223)(104 229 109 224)(105 230 110 225)(106 231 111 226)(107 232 112 227)(108 233 113 228)
(1 132)(2 131)(3 130)(4 129)(5 128)(6 127)(7 121)(8 126)(9 125)(10 124)(11 123)(12 122)(13 135)(14 134)(15 133)(16 138)(17 137)(18 136)(19 143)(20 142)(21 141)(22 140)(23 139)(24 144)(25 147)(26 146)(27 145)(28 150)(29 149)(30 148)(31 155)(32 154)(33 153)(34 152)(35 151)(36 156)(37 159)(38 158)(39 157)(40 162)(41 161)(42 160)(43 167)(44 166)(45 165)(46 164)(47 163)(48 168)(49 171)(50 170)(51 169)(52 174)(53 173)(54 172)(55 179)(56 178)(57 177)(58 176)(59 175)(60 180)(61 183)(62 182)(63 181)(64 186)(65 185)(66 184)(67 191)(68 190)(69 189)(70 188)(71 187)(72 192)(73 195)(74 194)(75 193)(76 198)(77 197)(78 196)(79 203)(80 202)(81 201)(82 200)(83 199)(84 204)(85 207)(86 206)(87 205)(88 210)(89 209)(90 208)(91 215)(92 214)(93 213)(94 212)(95 211)(96 216)(97 219)(98 218)(99 217)(100 222)(101 221)(102 220)(103 227)(104 226)(105 225)(106 224)(107 223)(108 228)(109 231)(110 230)(111 229)(112 234)(113 233)(114 232)(115 239)(116 238)(117 237)(118 236)(119 235)(120 240)

G:=sub<Sym(240)| (1,60,48,36,24)(2,55,43,31,19)(3,56,44,32,20)(4,57,45,33,21)(5,58,46,34,22)(6,59,47,35,23)(7,231,219,207,195)(8,232,220,208,196)(9,233,221,209,197)(10,234,222,210,198)(11,229,217,205,193)(12,230,218,206,194)(13,61,49,37,25)(14,62,50,38,26)(15,63,51,39,27)(16,64,52,40,28)(17,65,53,41,29)(18,66,54,42,30)(67,115,103,91,79)(68,116,104,92,80)(69,117,105,93,81)(70,118,106,94,82)(71,119,107,95,83)(72,120,108,96,84)(73,121,109,97,85)(74,122,110,98,86)(75,123,111,99,87)(76,124,112,100,88)(77,125,113,101,89)(78,126,114,102,90)(127,175,163,151,139)(128,176,164,152,140)(129,177,165,153,141)(130,178,166,154,142)(131,179,167,155,143)(132,180,168,156,144)(133,181,169,157,145)(134,182,170,158,146)(135,183,171,159,147)(136,184,172,160,148)(137,185,173,161,149)(138,186,174,162,150)(187,235,223,211,199)(188,236,224,212,200)(189,237,225,213,201)(190,238,226,214,202)(191,239,227,215,203)(192,240,228,216,204), (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)(157,158,159,160,161,162)(163,164,165,166,167,168)(169,170,171,172,173,174)(175,176,177,178,179,180)(181,182,183,184,185,186)(187,188,189,190,191,192)(193,194,195,196,197,198)(199,200,201,202,203,204)(205,206,207,208,209,210)(211,212,213,214,215,216)(217,218,219,220,221,222)(223,224,225,226,227,228)(229,230,231,232,233,234)(235,236,237,238,239,240), (1,69,4,72)(2,68,5,71)(3,67,6,70)(7,181,10,184)(8,186,11,183)(9,185,12,182)(13,78,16,75)(14,77,17,74)(15,76,18,73)(19,80,22,83)(20,79,23,82)(21,84,24,81)(25,90,28,87)(26,89,29,86)(27,88,30,85)(31,92,34,95)(32,91,35,94)(33,96,36,93)(37,102,40,99)(38,101,41,98)(39,100,42,97)(43,104,46,107)(44,103,47,106)(45,108,48,105)(49,114,52,111)(50,113,53,110)(51,112,54,109)(55,116,58,119)(56,115,59,118)(57,120,60,117)(61,126,64,123)(62,125,65,122)(63,124,66,121)(127,188,130,191)(128,187,131,190)(129,192,132,189)(133,198,136,195)(134,197,137,194)(135,196,138,193)(139,200,142,203)(140,199,143,202)(141,204,144,201)(145,210,148,207)(146,209,149,206)(147,208,150,205)(151,212,154,215)(152,211,155,214)(153,216,156,213)(157,222,160,219)(158,221,161,218)(159,220,162,217)(163,224,166,227)(164,223,167,226)(165,228,168,225)(169,234,172,231)(170,233,173,230)(171,232,174,229)(175,236,178,239)(176,235,179,238)(177,240,180,237), (1,137,17,132)(2,138,18,127)(3,133,13,128)(4,134,14,129)(5,135,15,130)(6,136,16,131)(7,123,238,118)(8,124,239,119)(9,125,240,120)(10,126,235,115)(11,121,236,116)(12,122,237,117)(19,150,30,139)(20,145,25,140)(21,146,26,141)(22,147,27,142)(23,148,28,143)(24,149,29,144)(31,162,42,151)(32,157,37,152)(33,158,38,153)(34,159,39,154)(35,160,40,155)(36,161,41,156)(43,174,54,163)(44,169,49,164)(45,170,50,165)(46,171,51,166)(47,172,52,167)(48,173,53,168)(55,186,66,175)(56,181,61,176)(57,182,62,177)(58,183,63,178)(59,184,64,179)(60,185,65,180)(67,198,78,187)(68,193,73,188)(69,194,74,189)(70,195,75,190)(71,196,76,191)(72,197,77,192)(79,210,90,199)(80,205,85,200)(81,206,86,201)(82,207,87,202)(83,208,88,203)(84,209,89,204)(91,222,102,211)(92,217,97,212)(93,218,98,213)(94,219,99,214)(95,220,100,215)(96,221,101,216)(103,234,114,223)(104,229,109,224)(105,230,110,225)(106,231,111,226)(107,232,112,227)(108,233,113,228), (1,132)(2,131)(3,130)(4,129)(5,128)(6,127)(7,121)(8,126)(9,125)(10,124)(11,123)(12,122)(13,135)(14,134)(15,133)(16,138)(17,137)(18,136)(19,143)(20,142)(21,141)(22,140)(23,139)(24,144)(25,147)(26,146)(27,145)(28,150)(29,149)(30,148)(31,155)(32,154)(33,153)(34,152)(35,151)(36,156)(37,159)(38,158)(39,157)(40,162)(41,161)(42,160)(43,167)(44,166)(45,165)(46,164)(47,163)(48,168)(49,171)(50,170)(51,169)(52,174)(53,173)(54,172)(55,179)(56,178)(57,177)(58,176)(59,175)(60,180)(61,183)(62,182)(63,181)(64,186)(65,185)(66,184)(67,191)(68,190)(69,189)(70,188)(71,187)(72,192)(73,195)(74,194)(75,193)(76,198)(77,197)(78,196)(79,203)(80,202)(81,201)(82,200)(83,199)(84,204)(85,207)(86,206)(87,205)(88,210)(89,209)(90,208)(91,215)(92,214)(93,213)(94,212)(95,211)(96,216)(97,219)(98,218)(99,217)(100,222)(101,221)(102,220)(103,227)(104,226)(105,225)(106,224)(107,223)(108,228)(109,231)(110,230)(111,229)(112,234)(113,233)(114,232)(115,239)(116,238)(117,237)(118,236)(119,235)(120,240)>;

G:=Group( (1,60,48,36,24)(2,55,43,31,19)(3,56,44,32,20)(4,57,45,33,21)(5,58,46,34,22)(6,59,47,35,23)(7,231,219,207,195)(8,232,220,208,196)(9,233,221,209,197)(10,234,222,210,198)(11,229,217,205,193)(12,230,218,206,194)(13,61,49,37,25)(14,62,50,38,26)(15,63,51,39,27)(16,64,52,40,28)(17,65,53,41,29)(18,66,54,42,30)(67,115,103,91,79)(68,116,104,92,80)(69,117,105,93,81)(70,118,106,94,82)(71,119,107,95,83)(72,120,108,96,84)(73,121,109,97,85)(74,122,110,98,86)(75,123,111,99,87)(76,124,112,100,88)(77,125,113,101,89)(78,126,114,102,90)(127,175,163,151,139)(128,176,164,152,140)(129,177,165,153,141)(130,178,166,154,142)(131,179,167,155,143)(132,180,168,156,144)(133,181,169,157,145)(134,182,170,158,146)(135,183,171,159,147)(136,184,172,160,148)(137,185,173,161,149)(138,186,174,162,150)(187,235,223,211,199)(188,236,224,212,200)(189,237,225,213,201)(190,238,226,214,202)(191,239,227,215,203)(192,240,228,216,204), (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)(157,158,159,160,161,162)(163,164,165,166,167,168)(169,170,171,172,173,174)(175,176,177,178,179,180)(181,182,183,184,185,186)(187,188,189,190,191,192)(193,194,195,196,197,198)(199,200,201,202,203,204)(205,206,207,208,209,210)(211,212,213,214,215,216)(217,218,219,220,221,222)(223,224,225,226,227,228)(229,230,231,232,233,234)(235,236,237,238,239,240), (1,69,4,72)(2,68,5,71)(3,67,6,70)(7,181,10,184)(8,186,11,183)(9,185,12,182)(13,78,16,75)(14,77,17,74)(15,76,18,73)(19,80,22,83)(20,79,23,82)(21,84,24,81)(25,90,28,87)(26,89,29,86)(27,88,30,85)(31,92,34,95)(32,91,35,94)(33,96,36,93)(37,102,40,99)(38,101,41,98)(39,100,42,97)(43,104,46,107)(44,103,47,106)(45,108,48,105)(49,114,52,111)(50,113,53,110)(51,112,54,109)(55,116,58,119)(56,115,59,118)(57,120,60,117)(61,126,64,123)(62,125,65,122)(63,124,66,121)(127,188,130,191)(128,187,131,190)(129,192,132,189)(133,198,136,195)(134,197,137,194)(135,196,138,193)(139,200,142,203)(140,199,143,202)(141,204,144,201)(145,210,148,207)(146,209,149,206)(147,208,150,205)(151,212,154,215)(152,211,155,214)(153,216,156,213)(157,222,160,219)(158,221,161,218)(159,220,162,217)(163,224,166,227)(164,223,167,226)(165,228,168,225)(169,234,172,231)(170,233,173,230)(171,232,174,229)(175,236,178,239)(176,235,179,238)(177,240,180,237), (1,137,17,132)(2,138,18,127)(3,133,13,128)(4,134,14,129)(5,135,15,130)(6,136,16,131)(7,123,238,118)(8,124,239,119)(9,125,240,120)(10,126,235,115)(11,121,236,116)(12,122,237,117)(19,150,30,139)(20,145,25,140)(21,146,26,141)(22,147,27,142)(23,148,28,143)(24,149,29,144)(31,162,42,151)(32,157,37,152)(33,158,38,153)(34,159,39,154)(35,160,40,155)(36,161,41,156)(43,174,54,163)(44,169,49,164)(45,170,50,165)(46,171,51,166)(47,172,52,167)(48,173,53,168)(55,186,66,175)(56,181,61,176)(57,182,62,177)(58,183,63,178)(59,184,64,179)(60,185,65,180)(67,198,78,187)(68,193,73,188)(69,194,74,189)(70,195,75,190)(71,196,76,191)(72,197,77,192)(79,210,90,199)(80,205,85,200)(81,206,86,201)(82,207,87,202)(83,208,88,203)(84,209,89,204)(91,222,102,211)(92,217,97,212)(93,218,98,213)(94,219,99,214)(95,220,100,215)(96,221,101,216)(103,234,114,223)(104,229,109,224)(105,230,110,225)(106,231,111,226)(107,232,112,227)(108,233,113,228), (1,132)(2,131)(3,130)(4,129)(5,128)(6,127)(7,121)(8,126)(9,125)(10,124)(11,123)(12,122)(13,135)(14,134)(15,133)(16,138)(17,137)(18,136)(19,143)(20,142)(21,141)(22,140)(23,139)(24,144)(25,147)(26,146)(27,145)(28,150)(29,149)(30,148)(31,155)(32,154)(33,153)(34,152)(35,151)(36,156)(37,159)(38,158)(39,157)(40,162)(41,161)(42,160)(43,167)(44,166)(45,165)(46,164)(47,163)(48,168)(49,171)(50,170)(51,169)(52,174)(53,173)(54,172)(55,179)(56,178)(57,177)(58,176)(59,175)(60,180)(61,183)(62,182)(63,181)(64,186)(65,185)(66,184)(67,191)(68,190)(69,189)(70,188)(71,187)(72,192)(73,195)(74,194)(75,193)(76,198)(77,197)(78,196)(79,203)(80,202)(81,201)(82,200)(83,199)(84,204)(85,207)(86,206)(87,205)(88,210)(89,209)(90,208)(91,215)(92,214)(93,213)(94,212)(95,211)(96,216)(97,219)(98,218)(99,217)(100,222)(101,221)(102,220)(103,227)(104,226)(105,225)(106,224)(107,223)(108,228)(109,231)(110,230)(111,229)(112,234)(113,233)(114,232)(115,239)(116,238)(117,237)(118,236)(119,235)(120,240) );

G=PermutationGroup([(1,60,48,36,24),(2,55,43,31,19),(3,56,44,32,20),(4,57,45,33,21),(5,58,46,34,22),(6,59,47,35,23),(7,231,219,207,195),(8,232,220,208,196),(9,233,221,209,197),(10,234,222,210,198),(11,229,217,205,193),(12,230,218,206,194),(13,61,49,37,25),(14,62,50,38,26),(15,63,51,39,27),(16,64,52,40,28),(17,65,53,41,29),(18,66,54,42,30),(67,115,103,91,79),(68,116,104,92,80),(69,117,105,93,81),(70,118,106,94,82),(71,119,107,95,83),(72,120,108,96,84),(73,121,109,97,85),(74,122,110,98,86),(75,123,111,99,87),(76,124,112,100,88),(77,125,113,101,89),(78,126,114,102,90),(127,175,163,151,139),(128,176,164,152,140),(129,177,165,153,141),(130,178,166,154,142),(131,179,167,155,143),(132,180,168,156,144),(133,181,169,157,145),(134,182,170,158,146),(135,183,171,159,147),(136,184,172,160,148),(137,185,173,161,149),(138,186,174,162,150),(187,235,223,211,199),(188,236,224,212,200),(189,237,225,213,201),(190,238,226,214,202),(191,239,227,215,203),(192,240,228,216,204)], [(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),(157,158,159,160,161,162),(163,164,165,166,167,168),(169,170,171,172,173,174),(175,176,177,178,179,180),(181,182,183,184,185,186),(187,188,189,190,191,192),(193,194,195,196,197,198),(199,200,201,202,203,204),(205,206,207,208,209,210),(211,212,213,214,215,216),(217,218,219,220,221,222),(223,224,225,226,227,228),(229,230,231,232,233,234),(235,236,237,238,239,240)], [(1,69,4,72),(2,68,5,71),(3,67,6,70),(7,181,10,184),(8,186,11,183),(9,185,12,182),(13,78,16,75),(14,77,17,74),(15,76,18,73),(19,80,22,83),(20,79,23,82),(21,84,24,81),(25,90,28,87),(26,89,29,86),(27,88,30,85),(31,92,34,95),(32,91,35,94),(33,96,36,93),(37,102,40,99),(38,101,41,98),(39,100,42,97),(43,104,46,107),(44,103,47,106),(45,108,48,105),(49,114,52,111),(50,113,53,110),(51,112,54,109),(55,116,58,119),(56,115,59,118),(57,120,60,117),(61,126,64,123),(62,125,65,122),(63,124,66,121),(127,188,130,191),(128,187,131,190),(129,192,132,189),(133,198,136,195),(134,197,137,194),(135,196,138,193),(139,200,142,203),(140,199,143,202),(141,204,144,201),(145,210,148,207),(146,209,149,206),(147,208,150,205),(151,212,154,215),(152,211,155,214),(153,216,156,213),(157,222,160,219),(158,221,161,218),(159,220,162,217),(163,224,166,227),(164,223,167,226),(165,228,168,225),(169,234,172,231),(170,233,173,230),(171,232,174,229),(175,236,178,239),(176,235,179,238),(177,240,180,237)], [(1,137,17,132),(2,138,18,127),(3,133,13,128),(4,134,14,129),(5,135,15,130),(6,136,16,131),(7,123,238,118),(8,124,239,119),(9,125,240,120),(10,126,235,115),(11,121,236,116),(12,122,237,117),(19,150,30,139),(20,145,25,140),(21,146,26,141),(22,147,27,142),(23,148,28,143),(24,149,29,144),(31,162,42,151),(32,157,37,152),(33,158,38,153),(34,159,39,154),(35,160,40,155),(36,161,41,156),(43,174,54,163),(44,169,49,164),(45,170,50,165),(46,171,51,166),(47,172,52,167),(48,173,53,168),(55,186,66,175),(56,181,61,176),(57,182,62,177),(58,183,63,178),(59,184,64,179),(60,185,65,180),(67,198,78,187),(68,193,73,188),(69,194,74,189),(70,195,75,190),(71,196,76,191),(72,197,77,192),(79,210,90,199),(80,205,85,200),(81,206,86,201),(82,207,87,202),(83,208,88,203),(84,209,89,204),(91,222,102,211),(92,217,97,212),(93,218,98,213),(94,219,99,214),(95,220,100,215),(96,221,101,216),(103,234,114,223),(104,229,109,224),(105,230,110,225),(106,231,111,226),(107,232,112,227),(108,233,113,228)], [(1,132),(2,131),(3,130),(4,129),(5,128),(6,127),(7,121),(8,126),(9,125),(10,124),(11,123),(12,122),(13,135),(14,134),(15,133),(16,138),(17,137),(18,136),(19,143),(20,142),(21,141),(22,140),(23,139),(24,144),(25,147),(26,146),(27,145),(28,150),(29,149),(30,148),(31,155),(32,154),(33,153),(34,152),(35,151),(36,156),(37,159),(38,158),(39,157),(40,162),(41,161),(42,160),(43,167),(44,166),(45,165),(46,164),(47,163),(48,168),(49,171),(50,170),(51,169),(52,174),(53,173),(54,172),(55,179),(56,178),(57,177),(58,176),(59,175),(60,180),(61,183),(62,182),(63,181),(64,186),(65,185),(66,184),(67,191),(68,190),(69,189),(70,188),(71,187),(72,192),(73,195),(74,194),(75,193),(76,198),(77,197),(78,196),(79,203),(80,202),(81,201),(82,200),(83,199),(84,204),(85,207),(86,206),(87,205),(88,210),(89,209),(90,208),(91,215),(92,214),(93,213),(94,212),(95,211),(96,216),(97,219),(98,218),(99,217),(100,222),(101,221),(102,220),(103,227),(104,226),(105,225),(106,224),(107,223),(108,228),(109,231),(110,230),(111,229),(112,234),(113,233),(114,232),(115,239),(116,238),(117,237),(118,236),(119,235),(120,240)])

150 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A···4F4G4H4I4J4K4L5A5B5C5D6A6B6C10A···10L10M···10AB12A···12F15A15B15C15D20A···20X20Y···20AN20AO···20AV30A···30L60A···60X
order1222222234···4444444555566610···1010···1012···121515151520···2020···2020···2030···3060···60
size1111666622···233336611112221···16···64···422222···23···36···62···24···4

150 irreducible representations

dim1111111111111122222222224444
type+++++++++++
imageC1C2C2C2C2C2C4C5C10C10C10C10C10C20S3D4D6C4○D4C4×S3C5×S3C5×D4S3×C10C5×C4○D4S3×C20S3×D4Q83S3C5×S3×D4C5×Q83S3
kernelC5×Dic35D4Dic3×C20C5×D6⋊C4C15×C4⋊C4S3×C2×C20C10×D12C5×D12Dic35D4C4×Dic3D6⋊C4C3×C4⋊C4S3×C2×C4C2×D12D12C5×C4⋊C4C5×Dic3C2×C20C30C20C4⋊C4Dic3C2×C4C6C4C10C10C2C2
# reps1121218448484321232448128161144

Matrix representation of C5×Dic35D4 in GL5(𝔽61)

90000
09000
00900
00090
00009
,
600000
006000
016000
00010
00001
,
110000
00100
01000
000600
000060
,
10000
01000
00100
000159
000160
,
600000
00100
01000
000159
000060

G:=sub<GL(5,GF(61))| [9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[60,0,0,0,0,0,0,1,0,0,0,60,60,0,0,0,0,0,1,0,0,0,0,0,1],[11,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,59,60],[60,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,59,60] >;

C5×Dic35D4 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_3\rtimes_5D_4
% in TeX

G:=Group("C5xDic3:5D4");
// GroupNames label

G:=SmallGroup(480,772);
// by ID

G=gap.SmallGroup(480,772);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,568,891,226,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^6=d^4=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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