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G = C5×Dic34D4order 480 = 25·3·5

Direct product of C5 and Dic34D4

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

Aliases: C5×Dic34D4, C3⋊D4⋊C20, C32(D4×C20), C1536(C4×D4), D6⋊C49C10, D62(C2×C20), C222(S3×C20), Dic34(C5×D4), C6.18(D4×C10), Dic3⋊C49C10, (C5×Dic3)⋊19D4, Dic31(C2×C20), C10.170(S3×D4), (C2×C20).271D6, C30.354(C2×D4), C6.7(C22×C20), (Dic3×C20)⋊29C2, (C4×Dic3)⋊11C10, C23.19(S3×C10), (C22×C10).89D6, C30.244(C4○D4), (C2×C30).401C23, C30.198(C22×C4), (C2×C60).415C22, (C22×Dic3)⋊1C10, C10.109(D42S3), (C22×C30).116C22, (C10×Dic3).215C22, C2.2(C5×S3×D4), (S3×C2×C4)⋊9C10, C2.9(S3×C2×C20), (S3×C2×C20)⋊25C2, (C2×C6)⋊2(C2×C20), (C5×C3⋊D4)⋊5C4, (C2×C30)⋊30(C2×C4), (C2×C10)⋊16(C4×S3), (C5×D6⋊C4)⋊31C2, C22⋊C47(C5×S3), C10.134(S3×C2×C4), C6.21(C5×C4○D4), (S3×C10)⋊24(C2×C4), (C3×C22⋊C4)⋊9C10, (C5×C22⋊C4)⋊15S3, (C2×C4).26(S3×C10), (Dic3×C2×C10)⋊12C2, C2.2(C5×D42S3), (C2×C3⋊D4).2C10, (C10×C3⋊D4).9C2, C22.14(S3×C2×C10), (C15×C22⋊C4)⋊25C2, (C2×C12).54(C2×C10), (C5×Dic3⋊C4)⋊31C2, (C5×Dic3)⋊17(C2×C4), (S3×C2×C10).107C22, (C2×C6).22(C22×C10), (C22×C6).11(C2×C10), (C22×S3).17(C2×C10), (C2×C10).335(C22×S3), (C2×Dic3).47(C2×C10), SmallGroup(480,760)

Series: Derived Chief Lower central Upper central

C1C6 — C5×Dic34D4
C1C3C6C2×C6C2×C30S3×C2×C10C10×C3⋊D4 — C5×Dic34D4
C3C6 — C5×Dic34D4
C1C2×C10C5×C22⋊C4

Generators and relations for C5×Dic34D4
 G = < a,b,c,d,e | a5=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 404 in 188 conjugacy classes, 86 normal (58 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×2], C22 [×6], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C10 [×3], C10 [×4], Dic3 [×4], Dic3, C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4 [×2], C2×D4, C20 [×7], C2×C10, C2×C10 [×2], C2×C10 [×6], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C22×S3, C22×C6, C5×S3 [×2], C30 [×3], C30 [×2], C4×D4, C2×C20 [×2], C2×C20 [×7], C5×D4 [×4], C22×C10, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C5×Dic3 [×4], C5×Dic3, C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×C20, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C22×C20 [×2], D4×C10, Dic34D4, S3×C20 [×2], C10×Dic3 [×3], C10×Dic3 [×2], C5×C3⋊D4 [×4], C2×C60 [×2], S3×C2×C10, C22×C30, D4×C20, Dic3×C20, C5×Dic3⋊C4, C5×D6⋊C4, C15×C22⋊C4, S3×C2×C20, Dic3×C2×C10, C10×C3⋊D4, C5×Dic34D4
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, D42S3, S3×C10 [×3], C22×C20, D4×C10, C5×C4○D4, Dic34D4, S3×C20 [×2], S3×C2×C10, D4×C20, S3×C2×C20, C5×S3×D4, C5×D42S3, C5×Dic34D4

Smallest permutation representation of C5×Dic34D4
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)(2 136 18 131)(3 135 13 130)(4 134 14 129)(5 133 15 128)(6 138 16 127)(7 121 238 116)(8 126 239 115)(9 125 240 120)(10 124 235 119)(11 123 236 118)(12 122 237 117)(19 148 30 143)(20 147 25 142)(21 146 26 141)(22 145 27 140)(23 150 28 139)(24 149 29 144)(31 160 42 155)(32 159 37 154)(33 158 38 153)(34 157 39 152)(35 162 40 151)(36 161 41 156)(43 172 54 167)(44 171 49 166)(45 170 50 165)(46 169 51 164)(47 174 52 163)(48 173 53 168)(55 184 66 179)(56 183 61 178)(57 182 62 177)(58 181 63 176)(59 186 64 175)(60 185 65 180)(67 196 78 191)(68 195 73 190)(69 194 74 189)(70 193 75 188)(71 198 76 187)(72 197 77 192)(79 208 90 203)(80 207 85 202)(81 206 86 201)(82 205 87 200)(83 210 88 199)(84 209 89 204)(91 220 102 215)(92 219 97 214)(93 218 98 213)(94 217 99 212)(95 222 100 211)(96 221 101 216)(103 232 114 227)(104 231 109 226)(105 230 110 225)(106 229 111 224)(107 234 112 223)(108 233 113 228)
(1 132)(2 127)(3 128)(4 129)(5 130)(6 131)(7 123)(8 124)(9 125)(10 126)(11 121)(12 122)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(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)(36 156)(37 157)(38 158)(39 159)(40 160)(41 161)(42 162)(43 163)(44 164)(45 165)(46 166)(47 167)(48 168)(49 169)(50 170)(51 171)(52 172)(53 173)(54 174)(55 175)(56 176)(57 177)(58 178)(59 179)(60 180)(61 181)(62 182)(63 183)(64 184)(65 185)(66 186)(67 187)(68 188)(69 189)(70 190)(71 191)(72 192)(73 193)(74 194)(75 195)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 217)(98 218)(99 219)(100 220)(101 221)(102 222)(103 223)(104 224)(105 225)(106 226)(107 227)(108 228)(109 229)(110 230)(111 231)(112 232)(113 233)(114 234)(115 235)(116 236)(117 237)(118 238)(119 239)(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)(2,136,18,131)(3,135,13,130)(4,134,14,129)(5,133,15,128)(6,138,16,127)(7,121,238,116)(8,126,239,115)(9,125,240,120)(10,124,235,119)(11,123,236,118)(12,122,237,117)(19,148,30,143)(20,147,25,142)(21,146,26,141)(22,145,27,140)(23,150,28,139)(24,149,29,144)(31,160,42,155)(32,159,37,154)(33,158,38,153)(34,157,39,152)(35,162,40,151)(36,161,41,156)(43,172,54,167)(44,171,49,166)(45,170,50,165)(46,169,51,164)(47,174,52,163)(48,173,53,168)(55,184,66,179)(56,183,61,178)(57,182,62,177)(58,181,63,176)(59,186,64,175)(60,185,65,180)(67,196,78,191)(68,195,73,190)(69,194,74,189)(70,193,75,188)(71,198,76,187)(72,197,77,192)(79,208,90,203)(80,207,85,202)(81,206,86,201)(82,205,87,200)(83,210,88,199)(84,209,89,204)(91,220,102,215)(92,219,97,214)(93,218,98,213)(94,217,99,212)(95,222,100,211)(96,221,101,216)(103,232,114,227)(104,231,109,226)(105,230,110,225)(106,229,111,224)(107,234,112,223)(108,233,113,228), (1,132)(2,127)(3,128)(4,129)(5,130)(6,131)(7,123)(8,124)(9,125)(10,126)(11,121)(12,122)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(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)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,169)(50,170)(51,171)(52,172)(53,173)(54,174)(55,175)(56,176)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(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)(2,136,18,131)(3,135,13,130)(4,134,14,129)(5,133,15,128)(6,138,16,127)(7,121,238,116)(8,126,239,115)(9,125,240,120)(10,124,235,119)(11,123,236,118)(12,122,237,117)(19,148,30,143)(20,147,25,142)(21,146,26,141)(22,145,27,140)(23,150,28,139)(24,149,29,144)(31,160,42,155)(32,159,37,154)(33,158,38,153)(34,157,39,152)(35,162,40,151)(36,161,41,156)(43,172,54,167)(44,171,49,166)(45,170,50,165)(46,169,51,164)(47,174,52,163)(48,173,53,168)(55,184,66,179)(56,183,61,178)(57,182,62,177)(58,181,63,176)(59,186,64,175)(60,185,65,180)(67,196,78,191)(68,195,73,190)(69,194,74,189)(70,193,75,188)(71,198,76,187)(72,197,77,192)(79,208,90,203)(80,207,85,202)(81,206,86,201)(82,205,87,200)(83,210,88,199)(84,209,89,204)(91,220,102,215)(92,219,97,214)(93,218,98,213)(94,217,99,212)(95,222,100,211)(96,221,101,216)(103,232,114,227)(104,231,109,226)(105,230,110,225)(106,229,111,224)(107,234,112,223)(108,233,113,228), (1,132)(2,127)(3,128)(4,129)(5,130)(6,131)(7,123)(8,124)(9,125)(10,126)(11,121)(12,122)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(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)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,169)(50,170)(51,171)(52,172)(53,173)(54,174)(55,175)(56,176)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(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),(2,136,18,131),(3,135,13,130),(4,134,14,129),(5,133,15,128),(6,138,16,127),(7,121,238,116),(8,126,239,115),(9,125,240,120),(10,124,235,119),(11,123,236,118),(12,122,237,117),(19,148,30,143),(20,147,25,142),(21,146,26,141),(22,145,27,140),(23,150,28,139),(24,149,29,144),(31,160,42,155),(32,159,37,154),(33,158,38,153),(34,157,39,152),(35,162,40,151),(36,161,41,156),(43,172,54,167),(44,171,49,166),(45,170,50,165),(46,169,51,164),(47,174,52,163),(48,173,53,168),(55,184,66,179),(56,183,61,178),(57,182,62,177),(58,181,63,176),(59,186,64,175),(60,185,65,180),(67,196,78,191),(68,195,73,190),(69,194,74,189),(70,193,75,188),(71,198,76,187),(72,197,77,192),(79,208,90,203),(80,207,85,202),(81,206,86,201),(82,205,87,200),(83,210,88,199),(84,209,89,204),(91,220,102,215),(92,219,97,214),(93,218,98,213),(94,217,99,212),(95,222,100,211),(96,221,101,216),(103,232,114,227),(104,231,109,226),(105,230,110,225),(106,229,111,224),(107,234,112,223),(108,233,113,228)], [(1,132),(2,127),(3,128),(4,129),(5,130),(6,131),(7,123),(8,124),(9,125),(10,126),(11,121),(12,122),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(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),(36,156),(37,157),(38,158),(39,159),(40,160),(41,161),(42,162),(43,163),(44,164),(45,165),(46,166),(47,167),(48,168),(49,169),(50,170),(51,171),(52,172),(53,173),(54,174),(55,175),(56,176),(57,177),(58,178),(59,179),(60,180),(61,181),(62,182),(63,183),(64,184),(65,185),(66,186),(67,187),(68,188),(69,189),(70,190),(71,191),(72,192),(73,193),(74,194),(75,195),(76,196),(77,197),(78,198),(79,199),(80,200),(81,201),(82,202),(83,203),(84,204),(85,205),(86,206),(87,207),(88,208),(89,209),(90,210),(91,211),(92,212),(93,213),(94,214),(95,215),(96,216),(97,217),(98,218),(99,219),(100,220),(101,221),(102,222),(103,223),(104,224),(105,225),(106,226),(107,227),(108,228),(109,229),(110,230),(111,231),(112,232),(113,233),(114,234),(115,235),(116,236),(117,237),(118,238),(119,239),(120,240)])

150 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B5C5D6A6B6C6D6E10A···10L10M···10T10U···10AB12A12B12C12D15A15B15C15D20A···20P20Q···20AF20AG···20AV30A···30L30M···30T60A···60P
order12222222344444444444455556666610···1010···1010···10121212121515151520···2020···2020···2030···3030···3060···60
size1111226622222333366661111222441···12···26···6444422222···23···36···62···24···44···4

150 irreducible representations

dim1111111111111111112222222222224444
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C4C5C10C10C10C10C10C10C10C20S3D4D6D6C4○D4C4×S3C5×S3C5×D4S3×C10S3×C10C5×C4○D4S3×C20S3×D4D42S3C5×S3×D4C5×D42S3
kernelC5×Dic34D4Dic3×C20C5×Dic3⋊C4C5×D6⋊C4C15×C22⋊C4S3×C2×C20Dic3×C2×C10C10×C3⋊D4C5×C3⋊D4Dic34D4C4×Dic3Dic3⋊C4D6⋊C4C3×C22⋊C4S3×C2×C4C22×Dic3C2×C3⋊D4C3⋊D4C5×C22⋊C4C5×Dic3C2×C20C22×C10C30C2×C10C22⋊C4Dic3C2×C4C23C6C22C10C10C2C2
# reps111111118444444443212212448848161144

Matrix representation of C5×Dic34D4 in GL4(𝔽61) generated by

34000
03400
00340
00034
,
1100
60000
00600
00060
,
11000
505000
00500
00050
,
1000
606000
00113
00050
,
1000
0100
00113
002150
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[1,60,0,0,1,0,0,0,0,0,60,0,0,0,0,60],[11,50,0,0,0,50,0,0,0,0,50,0,0,0,0,50],[1,60,0,0,0,60,0,0,0,0,11,0,0,0,3,50],[1,0,0,0,0,1,0,0,0,0,11,21,0,0,3,50] >;

C5×Dic34D4 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_3\rtimes_4D_4
% in TeX

G:=Group("C5xDic3:4D4");
// GroupNames label

G:=SmallGroup(480,760);
// by ID

G=gap.SmallGroup(480,760);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,891,226,15686]);
// 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=d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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