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## G = C3×D10⋊3Q8order 480 = 25·3·5

### Direct product of C3 and D10⋊3Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×D10⋊3Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — D5×C2×C12 — C3×D10⋊3Q8
 Lower central C5 — C2×C10 — C3×D10⋊3Q8
 Upper central C1 — C2×C6 — C6×Q8

Generators and relations for C3×D103Q8
G = < a,b,c,d,e | a3=b10=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede-1=d-1 >

Subgroups: 448 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 [×3], 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, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C3×C22⋊C4 [×2], C3×C4⋊C4 [×3], C22×C12, C6×Q8, C3×Dic5 [×3], C60 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C2×C4×D5, Q8×C10, C3×C22⋊Q8, D5×C12 [×2], C6×Dic5, C6×Dic5 [×2], C2×C60, C2×C60 [×2], Q8×C15 [×2], D5×C2×C6, D103Q8, C3×C10.D4 [×2], C3×C4⋊Dic5, C3×D10⋊C4 [×2], D5×C2×C12, Q8×C30, C3×D103Q8
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, C5⋊D4 [×2], C22×D5, C6×D4, C6×Q8, C3×C4○D4, C6×D5 [×3], Q8×D5, Q82D5, C2×C5⋊D4, C3×C22⋊Q8, C3×C5⋊D4 [×2], D5×C2×C6, D103Q8, C3×Q8×D5, C3×Q82D5, C6×C5⋊D4, C3×D103Q8

Smallest permutation representation of C3×D103Q8
On 240 points
Generators in S240
(1 52 32)(2 53 33)(3 54 34)(4 55 35)(5 56 36)(6 57 37)(7 58 38)(8 59 39)(9 60 40)(10 51 31)(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)(11 236)(12 235)(13 234)(14 233)(15 232)(16 231)(17 240)(18 239)(19 238)(20 237)(31 45)(32 44)(33 43)(34 42)(35 41)(36 50)(37 49)(38 48)(39 47)(40 46)(51 65)(52 64)(53 63)(54 62)(55 61)(56 70)(57 69)(58 68)(59 67)(60 66)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(91 110)(92 109)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)(111 130)(112 129)(113 128)(114 127)(115 126)(116 125)(117 124)(118 123)(119 122)(120 121)(131 145)(132 144)(133 143)(134 142)(135 141)(136 150)(137 149)(138 148)(139 147)(140 146)(151 165)(152 164)(153 163)(154 162)(155 161)(156 170)(157 169)(158 168)(159 167)(160 166)(171 185)(172 184)(173 183)(174 182)(175 181)(176 190)(177 189)(178 188)(179 187)(180 186)(191 210)(192 209)(193 208)(194 207)(195 206)(196 205)(197 204)(198 203)(199 202)(200 201)(211 230)(212 229)(213 228)(214 227)(215 226)(216 225)(217 224)(218 223)(219 222)(220 221)
(1 85 25 72)(2 86 26 73)(3 87 27 74)(4 88 28 75)(5 89 29 76)(6 90 30 77)(7 81 21 78)(8 82 22 79)(9 83 23 80)(10 84 24 71)(11 172 232 185)(12 173 233 186)(13 174 234 187)(14 175 235 188)(15 176 236 189)(16 177 237 190)(17 178 238 181)(18 179 239 182)(19 180 240 183)(20 171 231 184)(31 104 44 91)(32 105 45 92)(33 106 46 93)(34 107 47 94)(35 108 48 95)(36 109 49 96)(37 110 50 97)(38 101 41 98)(39 102 42 99)(40 103 43 100)(51 124 64 111)(52 125 65 112)(53 126 66 113)(54 127 67 114)(55 128 68 115)(56 129 69 116)(57 130 70 117)(58 121 61 118)(59 122 62 119)(60 123 63 120)(131 191 144 204)(132 192 145 205)(133 193 146 206)(134 194 147 207)(135 195 148 208)(136 196 149 209)(137 197 150 210)(138 198 141 201)(139 199 142 202)(140 200 143 203)(151 211 164 224)(152 212 165 225)(153 213 166 226)(154 214 167 227)(155 215 168 228)(156 216 169 229)(157 217 170 230)(158 218 161 221)(159 219 162 222)(160 220 163 223)
(1 145 25 132)(2 146 26 133)(3 147 27 134)(4 148 28 135)(5 149 29 136)(6 150 30 137)(7 141 21 138)(8 142 22 139)(9 143 23 140)(10 144 24 131)(11 125 232 112)(12 126 233 113)(13 127 234 114)(14 128 235 115)(15 129 236 116)(16 130 237 117)(17 121 238 118)(18 122 239 119)(19 123 240 120)(20 124 231 111)(31 164 44 151)(32 165 45 152)(33 166 46 153)(34 167 47 154)(35 168 48 155)(36 169 49 156)(37 170 50 157)(38 161 41 158)(39 162 42 159)(40 163 43 160)(51 184 64 171)(52 185 65 172)(53 186 66 173)(54 187 67 174)(55 188 68 175)(56 189 69 176)(57 190 70 177)(58 181 61 178)(59 182 62 179)(60 183 63 180)(71 204 84 191)(72 205 85 192)(73 206 86 193)(74 207 87 194)(75 208 88 195)(76 209 89 196)(77 210 90 197)(78 201 81 198)(79 202 82 199)(80 203 83 200)(91 224 104 211)(92 225 105 212)(93 226 106 213)(94 227 107 214)(95 228 108 215)(96 229 109 216)(97 230 110 217)(98 221 101 218)(99 222 102 219)(100 223 103 220)

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

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

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

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 C5⋊D4 C3×C4○D4 C6×D5 C3×C5⋊D4 Q8×D5 Q8⋊2D5 C3×Q8×D5 C3×Q8⋊2D5 kernel C3×D10⋊3Q8 C3×C10.D4 C3×C4⋊Dic5 C3×D10⋊C4 D5×C2×C12 Q8×C30 D10⋊3Q8 C10.D4 C4⋊Dic5 D10⋊C4 C2×C4×D5 Q8×C10 C60 C6×D5 C6×Q8 C30 C2×C12 C20 D10 C2×Q8 C12 C10 C2×C4 C4 C6 C6 C2 C2 # reps 1 2 1 2 1 1 2 4 2 4 2 2 2 2 2 2 6 4 4 4 8 4 12 16 2 2 4 4

Matrix representation of C3×D103Q8 in GL6(𝔽61)

 13 0 0 0 0 0 0 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 43 0 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 60 1 0 0 0 0 0 0 60 43 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 59 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 46 0 0 0 0 53 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 38 0 0 0 0 46 52

G:=sub<GL(6,GF(61))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,18,0,0,0,0,43,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,60,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,43,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,59,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,53,0,0,0,0,46,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,46,0,0,0,0,38,52] >;

C3×D103Q8 in GAP, Magma, Sage, TeX

C_3\times D_{10}\rtimes_3Q_8
% in TeX

G:=Group("C3xD10:3Q8");
// GroupNames label

G:=SmallGroup(480,739);
// by ID

G=gap.SmallGroup(480,739);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,555,268,18822]);
// 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=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^5*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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