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G = C3×Dic5.14D4order 480 = 25·3·5

Direct product of C3 and Dic5.14D4

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

Aliases: C3×Dic5.14D4, (C2×C30)⋊6Q8, C4⋊Dic52C6, C10.4(C6×Q8), (C2×C6)⋊4Dic10, C6.168(D4×D5), C10.14(C6×D4), C30.76(C2×Q8), (C2×Dic10)⋊2C6, C30.328(C2×D4), C10.D44C6, C1526(C22⋊Q8), C2.6(C6×Dic10), C23.16(C6×D5), C23.D5.2C6, (C6×Dic10)⋊18C2, (C2×C12).230D10, Dic5.14(C3×D4), (C3×Dic5).75D4, C6.44(C2×Dic10), C222(C3×Dic10), (C22×C6).72D10, C30.228(C4○D4), (C2×C60).261C22, (C2×C30).336C23, C6.107(D42D5), (C22×Dic5).5C6, (C22×C30).94C22, (C6×Dic5).234C22, C2.6(C3×D4×D5), C51(C3×C22⋊Q8), (C2×C10)⋊2(C3×Q8), (C2×C4).5(C6×D5), (C2×C20).1(C2×C6), C22.39(D5×C2×C6), (C3×C4⋊Dic5)⋊20C2, C10.19(C3×C4○D4), C2.6(C3×D42D5), (C5×C22⋊C4).1C6, C22⋊C4.1(C3×D5), (C3×C22⋊C4).4D5, (C2×C6×Dic5).11C2, (C15×C22⋊C4).4C2, (C2×Dic5).5(C2×C6), (C3×C23.D5).8C2, (C3×C10.D4)⋊15C2, (C22×C10).13(C2×C6), (C2×C10).19(C22×C6), (C2×C6).332(C22×D5), SmallGroup(480,671)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×Dic5.14D4
C1C5C10C2×C10C2×C30C6×Dic5C2×C6×Dic5 — C3×Dic5.14D4
C5C2×C10 — C3×Dic5.14D4
C1C2×C6C3×C22⋊C4

Generators and relations for C3×Dic5.14D4
 G = < a,b,c,d,e | a3=b10=d4=e2=1, c2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede=b5d-1 >

Subgroups: 416 in 148 conjugacy classes, 70 normal (58 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×2], C5, C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C10 [×3], C10 [×2], C12 [×7], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4, C22⋊C4, C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12 [×2], C2×C12 [×6], C3×Q8 [×2], C22×C6, C30 [×3], C30 [×2], C22⋊Q8, Dic10 [×2], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×2], C22×C10, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4 [×3], C22×C12, C6×Q8, C3×Dic5 [×2], C3×Dic5 [×3], C60 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C10.D4 [×2], C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, C3×C22⋊Q8, C3×Dic10 [×2], C6×Dic5 [×4], C6×Dic5 [×2], C2×C60 [×2], C22×C30, Dic5.14D4, C3×C10.D4 [×2], C3×C4⋊Dic5, C3×C23.D5, C15×C22⋊C4, C6×Dic10, C2×C6×Dic5, C3×Dic5.14D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], Q8 [×2], C23, D5, C2×C6 [×7], C2×D4, C2×Q8, C4○D4, D10 [×3], C3×D4 [×2], C3×Q8 [×2], C22×C6, C3×D5, C22⋊Q8, Dic10 [×2], C22×D5, C6×D4, C6×Q8, C3×C4○D4, C6×D5 [×3], C2×Dic10, D4×D5, D42D5, C3×C22⋊Q8, C3×Dic10 [×2], D5×C2×C6, Dic5.14D4, C6×Dic10, C3×D4×D5, C3×D42D5, C3×Dic5.14D4

Smallest permutation representation of C3×Dic5.14D4
On 240 points
Generators in S240
(1 54 34)(2 55 35)(3 56 36)(4 57 37)(5 58 38)(6 59 39)(7 60 40)(8 51 31)(9 52 32)(10 53 33)(11 228 208)(12 229 209)(13 230 210)(14 221 201)(15 222 202)(16 223 203)(17 224 204)(18 225 205)(19 226 206)(20 227 207)(21 61 41)(22 62 42)(23 63 43)(24 64 44)(25 65 45)(26 66 46)(27 67 47)(28 68 48)(29 69 49)(30 70 50)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)(81 124 101)(82 125 102)(83 126 103)(84 127 104)(85 128 105)(86 129 106)(87 130 107)(88 121 108)(89 122 109)(90 123 110)(131 171 151)(132 172 152)(133 173 153)(134 174 154)(135 175 155)(136 176 156)(137 177 157)(138 178 158)(139 179 159)(140 180 160)(141 181 161)(142 182 162)(143 183 163)(144 184 164)(145 185 165)(146 186 166)(147 187 167)(148 188 168)(149 189 169)(150 190 170)(191 231 211)(192 232 212)(193 233 213)(194 234 214)(195 235 215)(196 236 216)(197 237 217)(198 238 218)(199 239 219)(200 240 220)
(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)
(1 143 6 148)(2 142 7 147)(3 141 8 146)(4 150 9 145)(5 149 10 144)(11 114 16 119)(12 113 17 118)(13 112 18 117)(14 111 19 116)(15 120 20 115)(21 133 26 138)(22 132 27 137)(23 131 28 136)(24 140 29 135)(25 139 30 134)(31 166 36 161)(32 165 37 170)(33 164 38 169)(34 163 39 168)(35 162 40 167)(41 153 46 158)(42 152 47 157)(43 151 48 156)(44 160 49 155)(45 159 50 154)(51 186 56 181)(52 185 57 190)(53 184 58 189)(54 183 59 188)(55 182 60 187)(61 173 66 178)(62 172 67 177)(63 171 68 176)(64 180 69 175)(65 179 70 174)(71 206 76 201)(72 205 77 210)(73 204 78 209)(74 203 79 208)(75 202 80 207)(81 193 86 198)(82 192 87 197)(83 191 88 196)(84 200 89 195)(85 199 90 194)(91 226 96 221)(92 225 97 230)(93 224 98 229)(94 223 99 228)(95 222 100 227)(101 213 106 218)(102 212 107 217)(103 211 108 216)(104 220 109 215)(105 219 110 214)(121 236 126 231)(122 235 127 240)(123 234 128 239)(124 233 129 238)(125 232 130 237)
(1 79 22 87)(2 80 23 88)(3 71 24 89)(4 72 25 90)(5 73 26 81)(6 74 27 82)(7 75 28 83)(8 76 29 84)(9 77 30 85)(10 78 21 86)(11 177 237 188)(12 178 238 189)(13 179 239 190)(14 180 240 181)(15 171 231 182)(16 172 232 183)(17 173 233 184)(18 174 234 185)(19 175 235 186)(20 176 236 187)(31 96 49 104)(32 97 50 105)(33 98 41 106)(34 99 42 107)(35 100 43 108)(36 91 44 109)(37 92 45 110)(38 93 46 101)(39 94 47 102)(40 95 48 103)(51 116 69 127)(52 117 70 128)(53 118 61 129)(54 119 62 130)(55 120 63 121)(56 111 64 122)(57 112 65 123)(58 113 66 124)(59 114 67 125)(60 115 68 126)(131 191 142 202)(132 192 143 203)(133 193 144 204)(134 194 145 205)(135 195 146 206)(136 196 147 207)(137 197 148 208)(138 198 149 209)(139 199 150 210)(140 200 141 201)(151 211 162 222)(152 212 163 223)(153 213 164 224)(154 214 165 225)(155 215 166 226)(156 216 167 227)(157 217 168 228)(158 218 169 229)(159 219 170 230)(160 220 161 221)
(11 232)(12 233)(13 234)(14 235)(15 236)(16 237)(17 238)(18 239)(19 240)(20 231)(71 84)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 81)(79 82)(80 83)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 101)(99 102)(100 103)(111 127)(112 128)(113 129)(114 130)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)(191 207)(192 208)(193 209)(194 210)(195 201)(196 202)(197 203)(198 204)(199 205)(200 206)(211 227)(212 228)(213 229)(214 230)(215 221)(216 222)(217 223)(218 224)(219 225)(220 226)

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

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

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

102 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F4G4H5A5B6A···6F6G6H6I6J10A···10F10G10H10I10J12A12B12C12D12E···12L12M12N12O12P15A15B15C15D20A···20H30A···30L30M···30T60A···60P
order1222223344444444556···6666610···10101010101212121212···12121212121515151520···2030···3030···3060···60
size1111221144101010102020221···122222···24444444410···102020202022224···42···24···44···4

102 irreducible representations

dim11111111111111222222222222224444
type++++++++-+++-+-
imageC1C2C2C2C2C2C2C3C6C6C6C6C6C6D4Q8D5C4○D4D10D10C3×D4C3×Q8C3×D5Dic10C3×C4○D4C6×D5C6×D5C3×Dic10D4×D5D42D5C3×D4×D5C3×D42D5
kernelC3×Dic5.14D4C3×C10.D4C3×C4⋊Dic5C3×C23.D5C15×C22⋊C4C6×Dic10C2×C6×Dic5Dic5.14D4C10.D4C4⋊Dic5C23.D5C5×C22⋊C4C2×Dic10C22×Dic5C3×Dic5C2×C30C3×C22⋊C4C30C2×C12C22×C6Dic5C2×C10C22⋊C4C2×C6C10C2×C4C23C22C6C6C2C2
# reps121111124222222222424448484162244

Matrix representation of C3×Dic5.14D4 in GL4(𝔽61) generated by

13000
01300
00470
00047
,
18100
60000
0010
0001
,
235500
73800
00600
00060
,
36400
572500
0012
006060
,
1000
0100
0010
006060
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,47,0,0,0,0,47],[18,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[23,7,0,0,55,38,0,0,0,0,60,0,0,0,0,60],[36,57,0,0,4,25,0,0,0,0,1,60,0,0,2,60],[1,0,0,0,0,1,0,0,0,0,1,60,0,0,0,60] >;

C3×Dic5.14D4 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_5._{14}D_4
% in TeX

G:=Group("C3xDic5.14D4");
// GroupNames label

G:=SmallGroup(480,671);
// by ID

G=gap.SmallGroup(480,671);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,590,555,142,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^10=d^4=e^2=1,c^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^5*c,c*e=e*c,e*d*e=b^5*d^-1>;
// generators/relations

׿
×
𝔽