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## G = C22×C15⋊7D4order 480 = 25·3·5

### Direct product of C22 and C15⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C22×C15⋊7D4
 Chief series C1 — C5 — C15 — C30 — D30 — C22×D15 — C23×D15 — C22×C15⋊7D4
 Lower central C15 — C30 — C22×C15⋊7D4
 Upper central C1 — C23 — C24

Generators and relations for C22×C157D4
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=ece=c-1, ede=d-1 >

Subgroups: 2452 in 472 conjugacy classes, 159 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, Dic3, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C24, C24, Dic5, D10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, C22×C6, D15, C30, C30, C30, C22×D4, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C22×C10, C22×C10, C22×Dic3, C2×C3⋊D4, S3×C23, C23×C6, Dic15, D30, D30, C2×C30, C2×C30, C22×Dic5, C2×C5⋊D4, C23×D5, C23×C10, C22×C3⋊D4, C2×Dic15, C157D4, C22×D15, C22×D15, C22×C30, C22×C30, C22×C30, C22×C5⋊D4, C22×Dic15, C2×C157D4, C23×D15, C23×C30, C22×C157D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C24, D10, C3⋊D4, C22×S3, D15, C22×D4, C5⋊D4, C22×D5, C2×C3⋊D4, S3×C23, D30, C2×C5⋊D4, C23×D5, C22×C3⋊D4, C157D4, C22×D15, C22×C5⋊D4, C2×C157D4, C23×D15, C22×C157D4

Smallest permutation representation of C22×C157D4
On 240 points
Generators in S240
(1 137)(2 138)(3 139)(4 140)(5 141)(6 142)(7 143)(8 144)(9 145)(10 146)(11 147)(12 148)(13 149)(14 150)(15 136)(16 132)(17 133)(18 134)(19 135)(20 121)(21 122)(22 123)(23 124)(24 125)(25 126)(26 127)(27 128)(28 129)(29 130)(30 131)(31 179)(32 180)(33 166)(34 167)(35 168)(36 169)(37 170)(38 171)(39 172)(40 173)(41 174)(42 175)(43 176)(44 177)(45 178)(46 156)(47 157)(48 158)(49 159)(50 160)(51 161)(52 162)(53 163)(54 164)(55 165)(56 151)(57 152)(58 153)(59 154)(60 155)(61 196)(62 197)(63 198)(64 199)(65 200)(66 201)(67 202)(68 203)(69 204)(70 205)(71 206)(72 207)(73 208)(74 209)(75 210)(76 189)(77 190)(78 191)(79 192)(80 193)(81 194)(82 195)(83 181)(84 182)(85 183)(86 184)(87 185)(88 186)(89 187)(90 188)(91 233)(92 234)(93 235)(94 236)(95 237)(96 238)(97 239)(98 240)(99 226)(100 227)(101 228)(102 229)(103 230)(104 231)(105 232)(106 220)(107 221)(108 222)(109 223)(110 224)(111 225)(112 211)(113 212)(114 213)(115 214)(116 215)(117 216)(118 217)(119 218)(120 219)
(1 88)(2 89)(3 90)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 106)(44 107)(45 108)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)(57 103)(58 104)(59 105)(60 91)(121 209)(122 210)(123 196)(124 197)(125 198)(126 199)(127 200)(128 201)(129 202)(130 203)(131 204)(132 205)(133 206)(134 207)(135 208)(136 185)(137 186)(138 187)(139 188)(140 189)(141 190)(142 191)(143 192)(144 193)(145 194)(146 195)(147 181)(148 182)(149 183)(150 184)(151 229)(152 230)(153 231)(154 232)(155 233)(156 234)(157 235)(158 236)(159 237)(160 238)(161 239)(162 240)(163 226)(164 227)(165 228)(166 225)(167 211)(168 212)(169 213)(170 214)(171 215)(172 216)(173 217)(174 218)(175 219)(176 220)(177 221)(178 222)(179 223)(180 224)
(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)
(1 164 28 169)(2 163 29 168)(3 162 30 167)(4 161 16 166)(5 160 17 180)(6 159 18 179)(7 158 19 178)(8 157 20 177)(9 156 21 176)(10 155 22 175)(11 154 23 174)(12 153 24 173)(13 152 25 172)(14 151 26 171)(15 165 27 170)(31 142 49 134)(32 141 50 133)(33 140 51 132)(34 139 52 131)(35 138 53 130)(36 137 54 129)(37 136 55 128)(38 150 56 127)(39 149 57 126)(40 148 58 125)(41 147 59 124)(42 146 60 123)(43 145 46 122)(44 144 47 121)(45 143 48 135)(61 219 82 233)(62 218 83 232)(63 217 84 231)(64 216 85 230)(65 215 86 229)(66 214 87 228)(67 213 88 227)(68 212 89 226)(69 211 90 240)(70 225 76 239)(71 224 77 238)(72 223 78 237)(73 222 79 236)(74 221 80 235)(75 220 81 234)(91 196 120 195)(92 210 106 194)(93 209 107 193)(94 208 108 192)(95 207 109 191)(96 206 110 190)(97 205 111 189)(98 204 112 188)(99 203 113 187)(100 202 114 186)(101 201 115 185)(102 200 116 184)(103 199 117 183)(104 198 118 182)(105 197 119 181)
(1 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 75)(9 74)(10 73)(11 72)(12 71)(13 70)(14 69)(15 68)(16 85)(17 84)(18 83)(19 82)(20 81)(21 80)(22 79)(23 78)(24 77)(25 76)(26 90)(27 89)(28 88)(29 87)(30 86)(31 119)(32 118)(33 117)(34 116)(35 115)(36 114)(37 113)(38 112)(39 111)(40 110)(41 109)(42 108)(43 107)(44 106)(45 120)(46 93)(47 92)(48 91)(49 105)(50 104)(51 103)(52 102)(53 101)(54 100)(55 99)(56 98)(57 97)(58 96)(59 95)(60 94)(121 194)(122 193)(123 192)(124 191)(125 190)(126 189)(127 188)(128 187)(129 186)(130 185)(131 184)(132 183)(133 182)(134 181)(135 195)(136 203)(137 202)(138 201)(139 200)(140 199)(141 198)(142 197)(143 196)(144 210)(145 209)(146 208)(147 207)(148 206)(149 205)(150 204)(151 240)(152 239)(153 238)(154 237)(155 236)(156 235)(157 234)(158 233)(159 232)(160 231)(161 230)(162 229)(163 228)(164 227)(165 226)(166 216)(167 215)(168 214)(169 213)(170 212)(171 211)(172 225)(173 224)(174 223)(175 222)(176 221)(177 220)(178 219)(179 218)(180 217)

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

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

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

132 conjugacy classes

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

132 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D5 D6 D10 C3⋊D4 D15 C5⋊D4 D30 C15⋊7D4 kernel C22×C15⋊7D4 C22×Dic15 C2×C15⋊7D4 C23×D15 C23×C30 C23×C10 C2×C30 C23×C6 C22×C10 C22×C6 C2×C10 C24 C2×C6 C23 C22 # reps 1 1 12 1 1 1 4 2 7 14 8 4 16 28 32

Matrix representation of C22×C157D4 in GL6(𝔽61)

 60 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 0 0 0 0 0 0 60
,
 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
,
 5 38 0 0 0 0 23 53 0 0 0 0 0 0 60 60 0 0 0 0 1 0 0 0 0 0 0 0 45 47 0 0 0 0 22 23
,
 60 0 0 0 0 0 18 1 0 0 0 0 0 0 1 0 0 0 0 0 60 60 0 0 0 0 0 0 8 14 0 0 0 0 52 53
,
 60 0 0 0 0 0 18 1 0 0 0 0 0 0 1 0 0 0 0 0 60 60 0 0 0 0 0 0 43 1 0 0 0 0 43 18

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[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],[5,23,0,0,0,0,38,53,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,45,22,0,0,0,0,47,23],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,8,52,0,0,0,0,14,53],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,43,43,0,0,0,0,1,18] >;

C22×C157D4 in GAP, Magma, Sage, TeX

C_2^2\times C_{15}\rtimes_7D_4
% in TeX

G:=Group("C2^2xC15:7D4");
// GroupNames label

G:=SmallGroup(480,1179);
// by ID

G=gap.SmallGroup(480,1179);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,675,2693,18822]);
// 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=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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