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## G = C22×C15⋊D4order 480 = 25·3·5

### Direct product of C22 and C15⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C22×C15⋊D4
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C15⋊D4 — C2×C15⋊D4 — C22×C15⋊D4
 Lower central C15 — C30 — C22×C15⋊D4
 Upper central C1 — C23

Generators and relations for C22×C15⋊D4
G = < a,b,c,d,e | a2=b2=c15=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece=c4, ede=d-1 >

Subgroups: 2044 in 472 conjugacy classes, 148 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×7], C22 [×32], C5, S3 [×4], C6, C6 [×6], C6 [×4], C2×C4 [×6], D4 [×16], C23, C23 [×20], D5 [×4], C10, C10 [×6], C10 [×4], Dic3 [×4], D6 [×4], D6 [×12], C2×C6 [×7], C2×C6 [×16], C15, C22×C4, C2×D4 [×12], C24 [×2], Dic5 [×4], D10 [×4], D10 [×12], C2×C10 [×7], C2×C10 [×16], C2×Dic3 [×6], C3⋊D4 [×16], C22×S3 [×6], C22×S3 [×4], C22×C6, C22×C6 [×10], C5×S3 [×4], C3×D5 [×4], C30, C30 [×6], C22×D4, C2×Dic5 [×6], C5⋊D4 [×16], C22×D5 [×6], C22×D5 [×4], C22×C10, C22×C10 [×10], C22×Dic3, C2×C3⋊D4 [×12], S3×C23, C23×C6, Dic15 [×4], C6×D5 [×4], C6×D5 [×12], S3×C10 [×4], S3×C10 [×12], C2×C30 [×7], C22×Dic5, C2×C5⋊D4 [×12], C23×D5, C23×C10, C22×C3⋊D4, C15⋊D4 [×16], C2×Dic15 [×6], D5×C2×C6 [×6], D5×C2×C6 [×4], S3×C2×C10 [×6], S3×C2×C10 [×4], C22×C30, C22×C5⋊D4, C2×C15⋊D4 [×12], C22×Dic15, D5×C22×C6, S3×C22×C10, C22×C15⋊D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C3⋊D4 [×4], C22×S3 [×7], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×C3⋊D4 [×6], S3×C23, S3×D5, C2×C5⋊D4 [×6], C23×D5, C22×C3⋊D4, C15⋊D4 [×4], C2×S3×D5 [×3], C22×C5⋊D4, C2×C15⋊D4 [×6], C22×S3×D5, C22×C15⋊D4

Smallest permutation representation of C22×C15⋊D4
On 240 points
Generators in S240
(1 120)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 102)(17 103)(18 104)(19 105)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 101)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 76)(39 77)(40 78)(41 79)(42 80)(43 81)(44 82)(45 83)(46 75)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 73)(60 74)(121 226)(122 227)(123 228)(124 229)(125 230)(126 231)(127 232)(128 233)(129 234)(130 235)(131 236)(132 237)(133 238)(134 239)(135 240)(136 224)(137 225)(138 211)(139 212)(140 213)(141 214)(142 215)(143 216)(144 217)(145 218)(146 219)(147 220)(148 221)(149 222)(150 223)(151 204)(152 205)(153 206)(154 207)(155 208)(156 209)(157 210)(158 196)(159 197)(160 198)(161 199)(162 200)(163 201)(164 202)(165 203)(166 183)(167 184)(168 185)(169 186)(170 187)(171 188)(172 189)(173 190)(174 191)(175 192)(176 193)(177 194)(178 195)(179 181)(180 182)
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 46)(16 44)(17 45)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(61 120)(62 106)(63 107)(64 108)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)(73 117)(74 118)(75 119)(76 96)(77 97)(78 98)(79 99)(80 100)(81 101)(82 102)(83 103)(84 104)(85 105)(86 91)(87 92)(88 93)(89 94)(90 95)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 176)(128 177)(129 178)(130 179)(131 180)(132 166)(133 167)(134 168)(135 169)(136 160)(137 161)(138 162)(139 163)(140 164)(141 165)(142 151)(143 152)(144 153)(145 154)(146 155)(147 156)(148 157)(149 158)(150 159)(181 235)(182 236)(183 237)(184 238)(185 239)(186 240)(187 226)(188 227)(189 228)(190 229)(191 230)(192 231)(193 232)(194 233)(195 234)(196 222)(197 223)(198 224)(199 225)(200 211)(201 212)(202 213)(203 214)(204 215)(205 216)(206 217)(207 218)(208 219)(209 220)(210 221)
(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)
(1 128 29 149)(2 127 30 148)(3 126 16 147)(4 125 17 146)(5 124 18 145)(6 123 19 144)(7 122 20 143)(8 121 21 142)(9 135 22 141)(10 134 23 140)(11 133 24 139)(12 132 25 138)(13 131 26 137)(14 130 27 136)(15 129 28 150)(31 154 51 173)(32 153 52 172)(33 152 53 171)(34 151 54 170)(35 165 55 169)(36 164 56 168)(37 163 57 167)(38 162 58 166)(39 161 59 180)(40 160 60 179)(41 159 46 178)(42 158 47 177)(43 157 48 176)(44 156 49 175)(45 155 50 174)(61 194 80 196)(62 193 81 210)(63 192 82 209)(64 191 83 208)(65 190 84 207)(66 189 85 206)(67 188 86 205)(68 187 87 204)(69 186 88 203)(70 185 89 202)(71 184 90 201)(72 183 76 200)(73 182 77 199)(74 181 78 198)(75 195 79 197)(91 216 111 227)(92 215 112 226)(93 214 113 240)(94 213 114 239)(95 212 115 238)(96 211 116 237)(97 225 117 236)(98 224 118 235)(99 223 119 234)(100 222 120 233)(101 221 106 232)(102 220 107 231)(103 219 108 230)(104 218 109 229)(105 217 110 228)
(1 100)(2 104)(3 93)(4 97)(5 101)(6 105)(7 94)(8 98)(9 102)(10 91)(11 95)(12 99)(13 103)(14 92)(15 96)(16 113)(17 117)(18 106)(19 110)(20 114)(21 118)(22 107)(23 111)(24 115)(25 119)(26 108)(27 112)(28 116)(29 120)(30 109)(31 62)(32 66)(33 70)(34 74)(35 63)(36 67)(37 71)(38 75)(39 64)(40 68)(41 72)(42 61)(43 65)(44 69)(45 73)(46 76)(47 80)(48 84)(49 88)(50 77)(51 81)(52 85)(53 89)(54 78)(55 82)(56 86)(57 90)(58 79)(59 83)(60 87)(121 235)(122 239)(123 228)(124 232)(125 236)(126 240)(127 229)(128 233)(129 237)(130 226)(131 230)(132 234)(133 238)(134 227)(135 231)(136 215)(137 219)(138 223)(139 212)(140 216)(141 220)(142 224)(143 213)(144 217)(145 221)(146 225)(147 214)(148 218)(149 222)(150 211)(151 198)(152 202)(153 206)(154 210)(155 199)(156 203)(157 207)(158 196)(159 200)(160 204)(161 208)(162 197)(163 201)(164 205)(165 209)(166 195)(167 184)(168 188)(169 192)(170 181)(171 185)(172 189)(173 193)(174 182)(175 186)(176 190)(177 194)(178 183)(179 187)(180 191)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 5A 5B 6A ··· 6G 6H ··· 6O 10A ··· 10N 10O ··· 10AD 15A 15B 30A ··· 30N order 1 2 ··· 2 2 2 2 2 2 2 2 2 3 4 4 4 4 5 5 6 ··· 6 6 ··· 6 10 ··· 10 10 ··· 10 15 15 30 ··· 30 size 1 1 ··· 1 6 6 6 6 10 10 10 10 2 30 30 30 30 2 2 2 ··· 2 10 ··· 10 2 ··· 2 6 ··· 6 4 4 4 ··· 4

84 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 D4 D5 D6 D6 D10 D10 C3⋊D4 C5⋊D4 S3×D5 C15⋊D4 C2×S3×D5 kernel C22×C15⋊D4 C2×C15⋊D4 C22×Dic15 D5×C22×C6 S3×C22×C10 C23×D5 C2×C30 S3×C23 C22×D5 C22×C10 C22×S3 C22×C6 C2×C10 C2×C6 C23 C22 C22 # reps 1 12 1 1 1 1 4 2 6 1 12 2 8 16 2 8 6

Matrix representation of C22×C15⋊D4 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 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 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 60 18 0 0 0 0 43 18 0 0 0 0 0 0 47 0 0 0 0 0 0 13
,
 2 15 0 0 0 0 20 59 0 0 0 0 0 0 18 43 0 0 0 0 1 43 0 0 0 0 0 0 0 60 0 0 0 0 1 0
,
 60 0 0 0 0 0 45 1 0 0 0 0 0 0 18 43 0 0 0 0 1 43 0 0 0 0 0 0 1 0 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,43,0,0,0,0,18,18,0,0,0,0,0,0,47,0,0,0,0,0,0,13],[2,20,0,0,0,0,15,59,0,0,0,0,0,0,18,1,0,0,0,0,43,43,0,0,0,0,0,0,0,1,0,0,0,0,60,0],[60,45,0,0,0,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,43,43,0,0,0,0,0,0,1,0,0,0,0,0,0,60] >;

C22×C15⋊D4 in GAP, Magma, Sage, TeX

C_2^2\times C_{15}\rtimes D_4
% in TeX

G:=Group("C2^2xC15:D4");
// GroupNames label

G:=SmallGroup(480,1118);
// by ID

G=gap.SmallGroup(480,1118);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,1356,18822]);
// Polycyclic

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

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