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## G = C3×D10.Q8order 480 = 25·3·5

### Direct product of C3 and D10.Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D10.Q8
 Chief series C1 — C5 — C10 — C20 — C4×D5 — D5×C12 — C3×C4.F5 — C3×D10.Q8
 Lower central C5 — C10 — C20 — C3×D10.Q8
 Upper central C1 — C6 — C12 — C24

Generators and relations for C3×D10.Q8
G = < a,b,c,d,e | a3=b10=c2=1, d4=b5, e2=b-1cd2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b3, cd=dc, ece-1=b7c, ede-1=b5d3 >

Subgroups: 184 in 60 conjugacy classes, 32 normal (28 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C2×C8, M4(2), Dic5, C20, D10, C24, C24, C2×C12, C3×D5, C30, C8.C4, C52C8, C40, C5⋊C8, C4×D5, C2×C24, C3×M4(2), C3×Dic5, C60, C6×D5, C8×D5, C4.F5, C3×C8.C4, C3×C52C8, C120, C3×C5⋊C8, D5×C12, D10.Q8, D5×C24, C3×C4.F5, C3×D10.Q8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C4⋊C4, F5, C2×C12, C3×D4, C3×Q8, C8.C4, C2×F5, C3×C4⋊C4, C3×F5, C4⋊F5, C3×C8.C4, C6×F5, D10.Q8, C3×C4⋊F5, C3×D10.Q8

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

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + - + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 Q8 C3×D4 C3×Q8 C8.C4 C3×C8.C4 F5 C2×F5 C3×F5 C4⋊F5 C6×F5 D10.Q8 C3×C4⋊F5 C3×D10.Q8 kernel C3×D10.Q8 D5×C24 C3×C4.F5 D10.Q8 C3×C5⋊2C8 C120 C8×D5 C4.F5 C5⋊2C8 C40 C3×Dic5 C6×D5 Dic5 D10 C15 C5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 2 2 4 4 4 1 1 2 2 4 8 1 1 2 2 2 4 4 8

Matrix representation of C3×D10.Q8 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 4 6 4 4 0 6 1 4 3 3 5 5 6 2 1 0
,
 6 3 4 2 6 5 1 1 1 1 5 6 3 6 1 5
,
 6 3 0 2 1 6 1 1 2 2 2 3 3 4 2 6
,
 6 0 4 1 5 6 5 2 2 0 4 1 2 4 6 5
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,0,3,6,6,6,3,2,4,1,5,1,4,4,5,0],[6,6,1,3,3,5,1,6,4,1,5,1,2,1,6,5],[6,1,2,3,3,6,2,4,0,1,2,2,2,1,3,6],[6,5,2,2,0,6,0,4,4,5,4,6,1,2,1,5] >;

C3×D10.Q8 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,276);
// by ID

G=gap.SmallGroup(480,276);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,512,136,2524,102,9414,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^10=c^2=1,d^4=b^5,e^2=b^-1*c*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,e*b*e^-1=b^3,c*d=d*c,e*c*e^-1=b^7*c,e*d*e^-1=b^5*d^3>;
// generators/relations

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