C2xC15:Q8 - GroupNames

G = C₂×C₁₅⋊Q₈ order 240 = 2⁴·3·5

Direct product of C₂ and C₁₅⋊Q₈

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

Aliases: C₂×C₁₅⋊Q₈, C₃₀⋊Q₈, C₁₀⋊₁Dic₆, C₆⋊₁Dic₁₀, C₃₀.₂₄C₂³, Dic₅.₁₄D₆, Dic₃.₉D₁₀, Dic₁₅.₁₅C₂², C₁₅⋊₄(C₂×Q₈), C₅⋊₂(C₂×Dic₆), C₃⋊₂(C₂×Dic₁₀), (C₂×C₆).₁₉D₁₀, (C₂×C₁₀).₁₉D₆, (C₆×Dic₅).₅C₂, (C₂×Dic₅).₄S₃, (C₂×Dic₃).₄D₅, C₂².₁₇(S₃×D₅), C₆.₂₄(C₂²×D₅), C₁₀.₂₄(C₂²×S₃), (C₂×C₃₀).₁₈C₂², (C₂×Dic₁₅).₈C₂, (C₁₀×Dic₃).₅C₂, (C₃×Dic₅).₁₆C₂², (C₅×Dic₃).₁₂C₂², C₂.₂₄(C₂×S₃×D₅), SmallGroup(240,148)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — C₂×C₁₅⋊Q₈

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₃×Dic₅ — C₁₅⋊Q₈ — C₂×C₁₅⋊Q₈

Lower central

C₁₅ — C₃₀ — C₂×C₁₅⋊Q₈

Upper central

C₁ — C₂²

Generators and relations for C₂×C₁₅⋊Q₈
G = < a,b,c,d | a²=b¹⁵=c⁴=1, d²=c², ab=ba, ac=ca, ad=da, cbc^-1=b¹¹, dbd^-1=b⁴, dcd^-1=c^-1 >

Subgroups: 272 in 76 conjugacy classes, 40 normal (18 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₅, C₆, C₆, C₂×C₄, Q₈, C₁₀, C₁₀, Dic₃, Dic₃, C₁₂, C₂×C₆, C₁₅, C₂×Q₈, Dic₅, Dic₅, C₂₀, C₂×C₁₀, Dic₆, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₃₀, C₃₀, Dic₁₀, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×Dic₆, C₅×Dic₃, C₃×Dic₅, Dic₁₅, C₂×C₃₀, C₂×Dic₁₀, C₁₅⋊Q₈, C₆×Dic₅, C₁₀×Dic₃, C₂×Dic₁₅, C₂×C₁₅⋊Q₈
Quotients: C₁, C₂, C₂², S₃, Q₈, C₂³, D₅, D₆, C₂×Q₈, D₁₀, Dic₆, C₂²×S₃, Dic₁₀, C₂²×D₅, C₂×Dic₆, S₃×D₅, C₂×Dic₁₀, C₁₅⋊Q₈, C₂×S₃×D₅, C₂×C₁₅⋊Q₈

Smallest permutation representation of C₂×C₁₅⋊Q₈
►Regular action on 240 points

Generators in S₂₄₀

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

(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)

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

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

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

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

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

C₂×C₁₅⋊Q₈ is a maximal subgroup of
Dic₅.₄D₁₂ Dic₅⋊₅Dic₆ Dic₃⋊₅Dic₁₀ Dic₁₅⋊₅Q₈ Dic₁₅⋊₁Q₈ Dic₃⋊Dic₁₀ Dic₁₅⋊Q₈ D₁₀⋊Dic₆ Dic₅.₈D₁₂ D₆⋊Dic₁₀ Dic₃.D₂₀ D₃₀⋊₈Q₈ D₆⋊₂Dic₁₀ D₃₀⋊₂Q₈ D₁₀⋊₂Dic₆ D₃₀⋊₄Q₈ Dic₁₅.D₄ D₆⋊₄Dic₁₀ Dic₁₅.₃₁D₄ C₆₀⋊Q₈ C₂₀⋊₄Dic₆ C₂₀⋊Dic₆ C₂³.D₅⋊S₃ Dic₁₅.₁₉D₄ C₆.(D₄×D₅) (C₂×C₃₀)⋊Q₈ (C₂×C₁₀)⋊₈Dic₆ Dic₁₅.₄₈D₄ C₂×D₅×Dic₆ C₂×S₃×Dic₁₀ C₁₅⋊2_-¹⁺⁴
C₂×C₁₅⋊Q₈ is a maximal quotient of
C₆₀.₆Q₈ C₁₂.Dic₁₀ C₂₀.Dic₆ C₆₀⋊Q₈ C₂₀⋊₄Dic₆ C₂₀⋊Dic₆ (C₂×C₃₀)⋊Q₈ (C₂×C₁₀)⋊₈Dic₆ Dic₁₅.₄₈D₄

42 conjugacy classes

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	4F	5A	5B	6A	6B	6C	10A	···	10F	12A	12B	12C	12D	15A	15B	20A	···	20H	30A	···	30F
order	1	2	2	2	3	4	4	4	4	4	4	5	5	6	6	6	10	···	10	12	12	12	12	15	15	20	···	20	30	···	30
size	1	1	1	1	2	6	6	10	10	30	30	2	2	2	2	2	2	···	2	10	10	10	10	4	4	6	···	6	4	···	4

42 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	-	+	+	+	+	+	-	-	+	-	+
image	C₁	C₂	C₂	C₂	C₂	S₃	Q₈	D₅	D₆	D₆	D₁₀	D₁₀	Dic₆	Dic₁₀	S₃×D₅	C₁₅⋊Q₈	C₂×S₃×D₅
kernel	C₂×C₁₅⋊Q₈	C₁₅⋊Q₈	C₆×Dic₅	C₁₀×Dic₃	C₂×Dic₁₅	C₂×Dic₅	C₃₀	C₂×Dic₃	Dic₅	C₂×C₁₀	Dic₃	C₂×C₆	C₁₀	C₆	C₂²	C₂	C₂
# reps	1	4	1	1	1	1	2	2	2	1	4	2	4	8	2	4	2

Matrix representation of C₂×C₁₅⋊Q₈ ►in GL₆(𝔽₆₁)

60	0	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	60	0
0	0	0	0	0	60

1	0	0	0	0	0
0	1	0	0	0	0
0	0	18	18	0	0
0	0	43	60	0	0
0	0	0	0	0	60
0	0	0	0	1	60

21	41	0	0	0	0
16	40	0	0	0	0
0	0	60	0	0	0
0	0	0	60	0	0
0	0	0	0	37	34
0	0	0	0	10	24

7	12	0	0	0	0
6	54	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

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,43,0,0,0,0,18,60,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[21,16,0,0,0,0,41,40,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,37,10,0,0,0,0,34,24],[7,6,0,0,0,0,12,54,0,0,0,0,0,0,60,0,0,0,0,0,43,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

C₂×C₁₅⋊Q₈ in GAP, Magma, Sage, TeX

C_2\times C_{15}\rtimes Q_8

% in TeX

G:=Group("C2xC15:Q8");

// GroupNames label

G:=SmallGroup(240,148);

// by ID

G=gap.SmallGroup(240,148);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,121,55,490,6917]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^15=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^11,d*b*d^-1=b^4,d*c*d^-1=c^-1>;

// generators/relations

G = C2×C15⋊Q8 order 240 = 24·3·5

Direct product of C2 and C15⋊Q8

G = C₂×C₁₅⋊Q₈ order 240 = 2⁴·3·5

Direct product of C₂ and C₁₅⋊Q₈