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## G = C3×D10⋊2Q8order 480 = 25·3·5

### Direct product of C3 and D10⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×D10⋊2Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — D5×C2×C12 — C3×D10⋊2Q8
 Lower central C5 — C2×C10 — C3×D10⋊2Q8
 Upper central C1 — C2×C6 — C3×C4⋊C4

Generators and relations for C3×D102Q8
G = < a,b,c,d,e | a3=b10=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=b3c, ce=ec, ede-1=d-1 >

Subgroups: 480 in 148 conjugacy classes, 70 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], C5, C6 [×3], C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, D5 [×2], C10 [×3], C12 [×2], C12 [×5], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C12, C2×C12 [×2], C2×C12 [×5], C3×Q8 [×2], C22×C6, C3×D5 [×2], C30 [×3], C22⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C3×C22⋊C4 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C22×C12, C6×Q8, C3×Dic5 [×3], C60 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C3×C22⋊Q8, C3×Dic10 [×2], D5×C12 [×2], C6×Dic5, C6×Dic5 [×2], C2×C60, C2×C60 [×2], D5×C2×C6, D102Q8, C3×C4⋊Dic5 [×2], C3×D10⋊C4 [×2], C15×C4⋊C4, C6×Dic10, D5×C2×C12, C3×D102Q8
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, D20 [×2], C22×D5, C6×D4, C6×Q8, C3×C4○D4, C6×D5 [×3], C2×D20, D42D5, Q8×D5, C3×C22⋊Q8, C3×D20 [×2], D5×C2×C6, D102Q8, C6×D20, C3×D42D5, C3×Q8×D5, C3×D102Q8

Smallest permutation representation of C3×D102Q8
On 240 points
Generators in S240
(1 60 40)(2 51 31)(3 52 32)(4 53 33)(5 54 34)(6 55 35)(7 56 36)(8 57 37)(9 58 38)(10 59 39)(11 225 205)(12 226 206)(13 227 207)(14 228 208)(15 229 209)(16 230 210)(17 221 201)(18 222 202)(19 223 203)(20 224 204)(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)(81 121 101)(82 122 102)(83 123 103)(84 124 104)(85 125 105)(86 126 106)(87 127 107)(88 128 108)(89 129 109)(90 130 110)(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 24)(2 23)(3 22)(4 21)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 234)(12 233)(13 232)(14 231)(15 240)(16 239)(17 238)(18 237)(19 236)(20 235)(31 43)(32 42)(33 41)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(51 63)(52 62)(53 61)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(71 88)(72 87)(73 86)(74 85)(75 84)(76 83)(77 82)(78 81)(79 90)(80 89)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(97 102)(98 101)(99 110)(100 109)(111 128)(112 127)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 130)(120 129)(131 143)(132 142)(133 141)(134 150)(135 149)(136 148)(137 147)(138 146)(139 145)(140 144)(151 163)(152 162)(153 161)(154 170)(155 169)(156 168)(157 167)(158 166)(159 165)(160 164)(171 183)(172 182)(173 181)(174 190)(175 189)(176 188)(177 187)(178 186)(179 185)(180 184)(191 208)(192 207)(193 206)(194 205)(195 204)(196 203)(197 202)(198 201)(199 210)(200 209)(211 228)(212 227)(213 226)(214 225)(215 224)(216 223)(217 222)(218 221)(219 230)(220 229)
(1 75 25 90)(2 74 26 89)(3 73 27 88)(4 72 28 87)(5 71 29 86)(6 80 30 85)(7 79 21 84)(8 78 22 83)(9 77 23 82)(10 76 24 81)(11 190 240 175)(12 189 231 174)(13 188 232 173)(14 187 233 172)(15 186 234 171)(16 185 235 180)(17 184 236 179)(18 183 237 178)(19 182 238 177)(20 181 239 176)(31 94 46 109)(32 93 47 108)(33 92 48 107)(34 91 49 106)(35 100 50 105)(36 99 41 104)(37 98 42 103)(38 97 43 102)(39 96 44 101)(40 95 45 110)(51 114 66 129)(52 113 67 128)(53 112 68 127)(54 111 69 126)(55 120 70 125)(56 119 61 124)(57 118 62 123)(58 117 63 122)(59 116 64 121)(60 115 65 130)(131 209 146 194)(132 208 147 193)(133 207 148 192)(134 206 149 191)(135 205 150 200)(136 204 141 199)(137 203 142 198)(138 202 143 197)(139 201 144 196)(140 210 145 195)(151 229 166 214)(152 228 167 213)(153 227 168 212)(154 226 169 211)(155 225 170 220)(156 224 161 219)(157 223 162 218)(158 222 163 217)(159 221 164 216)(160 230 165 215)
(1 145 25 140)(2 146 26 131)(3 147 27 132)(4 148 28 133)(5 149 29 134)(6 150 30 135)(7 141 21 136)(8 142 22 137)(9 143 23 138)(10 144 24 139)(11 125 240 120)(12 126 231 111)(13 127 232 112)(14 128 233 113)(15 129 234 114)(16 130 235 115)(17 121 236 116)(18 122 237 117)(19 123 238 118)(20 124 239 119)(31 166 46 151)(32 167 47 152)(33 168 48 153)(34 169 49 154)(35 170 50 155)(36 161 41 156)(37 162 42 157)(38 163 43 158)(39 164 44 159)(40 165 45 160)(51 186 66 171)(52 187 67 172)(53 188 68 173)(54 189 69 174)(55 190 70 175)(56 181 61 176)(57 182 62 177)(58 183 63 178)(59 184 64 179)(60 185 65 180)(71 206 86 191)(72 207 87 192)(73 208 88 193)(74 209 89 194)(75 210 90 195)(76 201 81 196)(77 202 82 197)(78 203 83 198)(79 204 84 199)(80 205 85 200)(91 226 106 211)(92 227 107 212)(93 228 108 213)(94 229 109 214)(95 230 110 215)(96 221 101 216)(97 222 102 217)(98 223 103 218)(99 224 104 219)(100 225 105 220)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A ··· 6F 6G 6H 6I 6J 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 12M 12N 12O 12P 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 5 5 6 ··· 6 6 6 6 6 10 ··· 10 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 1 1 2 2 4 4 10 10 20 20 2 2 1 ··· 1 10 10 10 10 2 ··· 2 2 2 2 2 4 4 4 4 10 10 10 10 20 20 20 20 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + - - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 Q8 D5 C4○D4 D10 C3×D4 C3×Q8 C3×D5 D20 C3×C4○D4 C6×D5 C3×D20 D4⋊2D5 Q8×D5 C3×D4⋊2D5 C3×Q8×D5 kernel C3×D10⋊2Q8 C3×C4⋊Dic5 C3×D10⋊C4 C15×C4⋊C4 C6×Dic10 D5×C2×C12 D10⋊2Q8 C4⋊Dic5 D10⋊C4 C5×C4⋊C4 C2×Dic10 C2×C4×D5 C60 C6×D5 C3×C4⋊C4 C30 C2×C12 C20 D10 C4⋊C4 C12 C10 C2×C4 C4 C6 C6 C2 C2 # reps 1 2 2 1 1 1 2 4 4 2 2 2 2 2 2 2 6 4 4 4 8 4 12 16 2 2 4 4

Matrix representation of C3×D102Q8 in GL4(𝔽61) generated by

 47 0 0 0 0 47 0 0 0 0 1 0 0 0 0 1
,
 1 44 0 0 17 17 0 0 0 0 1 0 0 0 0 1
,
 1 44 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 32 7 0 0 2 29 0 0 0 0 39 15 0 0 49 22
,
 60 0 0 0 0 60 0 0 0 0 17 8 0 0 40 44
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,1,0,0,0,0,1],[1,17,0,0,44,17,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,44,60,0,0,0,0,60,0,0,0,0,60],[32,2,0,0,7,29,0,0,0,0,39,49,0,0,15,22],[60,0,0,0,0,60,0,0,0,0,17,40,0,0,8,44] >;

C3×D102Q8 in GAP, Magma, Sage, TeX

C_3\times D_{10}\rtimes_2Q_8
% in TeX

G:=Group("C3xD10:2Q8");
// GroupNames label

G:=SmallGroup(480,690);
// by ID

G=gap.SmallGroup(480,690);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,555,394,18822]);
// Polycyclic

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

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