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## G = C3×Q8.D10order 480 = 25·3·5

### Direct product of C3 and Q8.D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×Q8.D10
 Chief series C1 — C5 — C10 — C20 — C60 — D5×C12 — C3×Q8⋊2D5 — C3×Q8.D10
 Lower central C5 — C10 — C20 — C3×Q8.D10
 Upper central C1 — C6 — C12 — C3×Q16

Generators and relations for C3×Q8.D10
G = < a,b,c,d,e | a3=b4=e2=1, c2=d10=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=b2d9 >

Subgroups: 464 in 124 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D4, Q8, D5, C10, C12, C12, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, D10, C24, C24, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C4○D8, C52C8, C40, C4×D5, C4×D5, D20, D20, C5×Q8, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C3×Dic5, C60, C60, C6×D5, C6×D5, C8×D5, D40, Q8⋊D5, C5×Q16, Q82D5, C3×C4○D8, C3×C52C8, C120, D5×C12, D5×C12, C3×D20, C3×D20, Q8×C15, Q8.D10, D5×C24, C3×D40, C3×Q8⋊D5, C15×Q16, C3×Q82D5, C3×Q8.D10
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C4○D8, C22×D5, C6×D4, C6×D5, D4×D5, C3×C4○D8, D5×C2×C6, Q8.D10, C3×D4×D5, C3×Q8.D10

Smallest permutation representation of C3×Q8.D10
On 240 points
Generators in S240
(1 38 161)(2 39 162)(3 40 163)(4 21 164)(5 22 165)(6 23 166)(7 24 167)(8 25 168)(9 26 169)(10 27 170)(11 28 171)(12 29 172)(13 30 173)(14 31 174)(15 32 175)(16 33 176)(17 34 177)(18 35 178)(19 36 179)(20 37 180)(41 235 207)(42 236 208)(43 237 209)(44 238 210)(45 239 211)(46 240 212)(47 221 213)(48 222 214)(49 223 215)(50 224 216)(51 225 217)(52 226 218)(53 227 219)(54 228 220)(55 229 201)(56 230 202)(57 231 203)(58 232 204)(59 233 205)(60 234 206)(61 92 139)(62 93 140)(63 94 121)(64 95 122)(65 96 123)(66 97 124)(67 98 125)(68 99 126)(69 100 127)(70 81 128)(71 82 129)(72 83 130)(73 84 131)(74 85 132)(75 86 133)(76 87 134)(77 88 135)(78 89 136)(79 90 137)(80 91 138)(101 150 193)(102 151 194)(103 152 195)(104 153 196)(105 154 197)(106 155 198)(107 156 199)(108 157 200)(109 158 181)(110 159 182)(111 160 183)(112 141 184)(113 142 185)(114 143 186)(115 144 187)(116 145 188)(117 146 189)(118 147 190)(119 148 191)(120 149 192)
(1 86 11 96)(2 97 12 87)(3 88 13 98)(4 99 14 89)(5 90 15 100)(6 81 16 91)(7 92 17 82)(8 83 18 93)(9 94 19 84)(10 85 20 95)(21 126 31 136)(22 137 32 127)(23 128 33 138)(24 139 34 129)(25 130 35 140)(26 121 36 131)(27 132 37 122)(28 123 38 133)(29 134 39 124)(30 125 40 135)(41 189 51 199)(42 200 52 190)(43 191 53 181)(44 182 54 192)(45 193 55 183)(46 184 56 194)(47 195 57 185)(48 186 58 196)(49 197 59 187)(50 188 60 198)(61 177 71 167)(62 168 72 178)(63 179 73 169)(64 170 74 180)(65 161 75 171)(66 172 76 162)(67 163 77 173)(68 174 78 164)(69 165 79 175)(70 176 80 166)(101 229 111 239)(102 240 112 230)(103 231 113 221)(104 222 114 232)(105 233 115 223)(106 224 116 234)(107 235 117 225)(108 226 118 236)(109 237 119 227)(110 228 120 238)(141 202 151 212)(142 213 152 203)(143 204 153 214)(144 215 154 205)(145 206 155 216)(146 217 156 207)(147 208 157 218)(148 219 158 209)(149 210 159 220)(150 201 160 211)
(1 51 11 41)(2 200 12 190)(3 53 13 43)(4 182 14 192)(5 55 15 45)(6 184 16 194)(7 57 17 47)(8 186 18 196)(9 59 19 49)(10 188 20 198)(21 110 31 120)(22 229 32 239)(23 112 33 102)(24 231 34 221)(25 114 35 104)(26 233 36 223)(27 116 37 106)(28 235 38 225)(29 118 39 108)(30 237 40 227)(42 87 52 97)(44 89 54 99)(46 91 56 81)(48 93 58 83)(50 95 60 85)(61 152 71 142)(62 204 72 214)(63 154 73 144)(64 206 74 216)(65 156 75 146)(66 208 76 218)(67 158 77 148)(68 210 78 220)(69 160 79 150)(70 212 80 202)(82 185 92 195)(84 187 94 197)(86 189 96 199)(88 191 98 181)(90 193 100 183)(101 127 111 137)(103 129 113 139)(105 131 115 121)(107 133 117 123)(109 135 119 125)(122 234 132 224)(124 236 134 226)(126 238 136 228)(128 240 138 230)(130 222 140 232)(141 176 151 166)(143 178 153 168)(145 180 155 170)(147 162 157 172)(149 164 159 174)(161 217 171 207)(163 219 173 209)(165 201 175 211)(167 203 177 213)(169 205 179 215)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 80)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 120)(114 119)(115 118)(116 117)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(135 140)(136 139)(137 138)(141 150)(142 149)(143 148)(144 147)(145 146)(151 160)(152 159)(153 158)(154 157)(155 156)(161 170)(162 169)(163 168)(164 167)(165 166)(171 180)(172 179)(173 178)(174 177)(175 176)(181 196)(182 195)(183 194)(184 193)(185 192)(186 191)(187 190)(188 189)(197 200)(198 199)(201 212)(202 211)(203 210)(204 209)(205 208)(206 207)(213 220)(214 219)(215 218)(216 217)(221 228)(222 227)(223 226)(224 225)(229 240)(230 239)(231 238)(232 237)(233 236)(234 235)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 10A 10B 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 24A 24B 24C 24D 24E 24F 24G 24H 30A 30B 30C 30D 40A 40B 40C 40D 60A 60B 60C 60D 60E ··· 60L 120A ··· 120H order 1 2 2 2 2 3 3 4 4 4 4 4 5 5 6 6 6 6 6 6 6 6 8 8 8 8 10 10 12 12 12 12 12 12 12 12 12 12 15 15 15 15 20 20 20 20 20 20 24 24 24 24 24 24 24 24 30 30 30 30 40 40 40 40 60 60 60 60 60 ··· 60 120 ··· 120 size 1 1 10 20 20 1 1 2 4 4 5 5 2 2 1 1 10 10 20 20 20 20 2 2 10 10 2 2 2 2 4 4 4 4 5 5 5 5 2 2 2 2 4 4 8 8 8 8 2 2 2 2 10 10 10 10 2 2 2 2 4 4 4 4 4 4 4 4 8 ··· 8 4 ··· 4

84 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 D4 D5 D10 D10 C3×D4 C3×D4 C3×D5 C4○D8 C6×D5 C6×D5 C3×C4○D8 D4×D5 Q8.D10 C3×D4×D5 C3×Q8.D10 kernel C3×Q8.D10 D5×C24 C3×D40 C3×Q8⋊D5 C15×Q16 C3×Q8⋊2D5 Q8.D10 C8×D5 D40 Q8⋊D5 C5×Q16 Q8⋊2D5 C3×Dic5 C6×D5 C3×Q16 C24 C3×Q8 Dic5 D10 Q16 C15 C8 Q8 C5 C6 C3 C2 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 1 2 2 4 2 2 4 4 4 8 8 2 4 4 8

Matrix representation of C3×Q8.D10 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 177 204 0 0 0 64
,
 240 0 0 0 0 240 0 0 0 0 176 140 0 0 49 65
,
 240 51 0 0 190 190 0 0 0 0 203 83 0 0 151 38
,
 240 51 0 0 0 1 0 0 0 0 203 8 0 0 151 38
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,177,0,0,0,204,64],[240,0,0,0,0,240,0,0,0,0,176,49,0,0,140,65],[240,190,0,0,51,190,0,0,0,0,203,151,0,0,83,38],[240,0,0,0,51,1,0,0,0,0,203,151,0,0,8,38] >;

C3×Q8.D10 in GAP, Magma, Sage, TeX

C_3\times Q_8.D_{10}
% in TeX

G:=Group("C3xQ8.D10");
// GroupNames label

G:=SmallGroup(480,712);
// by ID

G=gap.SmallGroup(480,712);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,344,1094,303,268,1271,648,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=e^2=1,c^2=d^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=b^2*d^9>;
// generators/relations

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