C2^3.11D30 - GroupNames

G = C₂³.₁₁D₃₀ order 480 = 2⁵·3·5

6^th non-split extension by C₂³ of D₃₀ acting via D₃₀/C₁₅=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₁₁D₃₀, Dic₁₅.₂₀D₄, (C₂×C₄).₈D₃₀, C₂²⋊C₄⋊₅D₁₅, C₆.₁₀₁(D₄×D₅), (C₂×C₂₀).₃₄D₆, C₂.₁₀(D₄×D₁₅), D₃₀⋊₃C₄⋊₆C₂, (C₂×Dic₃₀)⋊₆C₂, (C₂×C₁₂).₃₄D₁₀, C₃₀.₃₀₉(C₂×D₄), C₁₀.₁₀₃(S₃×D₄), (C₄×Dic₁₅)⋊₁₈C₂, C₆.₉₈(C₄○D₂₀), C₃₀.₃₈D₄⋊₅C₂, C₁₅⋊₁₉(C₄.₄D₄), (C₂²×C₆).₆₀D₁₀, (C₂²×C₁₀).₇₅D₆, C₁₀.₉₈(C₄○D₁₂), C₃₀.₁₇₁(C₄○D₄), C₆.₉₃(D₄⋊₂D₅), C₂.₉(D₄⋊₂D₁₅), (C₂×C₆₀).₁₇₅C₂², (C₂×C₃₀).₂₈₂C₂³, C₃⋊₅(Dic₅.₅D₄), C₅⋊₅(C₂³.₁₁D₆), C₁₀.₉₃(D₄⋊₂S₃), (C₂²×C₃₀).₁₆C₂², C₂.₁₂(D₆₀⋊₁₁C₂), (C₂×Dic₁₅).₈C₂², (C₂²×D₁₅).₅C₂², C₂².₄₄(C₂²×D₁₅), (C₅×C₂²⋊C₄)⋊₇S₃, (C₃×C₂²⋊C₄)⋊₇D₅, (C₁₅×C₂²⋊C₄)⋊₉C₂, (C₂×C₁₅⋊₇D₄).₄C₂, (C₂×C₆).₂₇₈(C₂²×D₅), (C₂×C₁₀).₂₇₇(C₂²×S₃), SmallGroup(480,850)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₃₀ — C₂³.₁₁D₃₀

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₂×C₃₀ — C₂²×D₁₅ — D₃₀⋊₃C₄ — C₂³.₁₁D₃₀

Lower central

C₁₅ — C₂×C₃₀ — C₂³.₁₁D₃₀

Upper central

C₁ — C₂² — C₂²⋊C₄

Generators and relations for C₂³.₁₁D₃₀
G = < a,b,c,d,e | a²=b²=c²=1, d³⁰=b, e²=cb=bc, eae^-1=ab=ba, dad^-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede^-1=cd²⁹ >

Subgroups: 932 in 152 conjugacy classes, 49 normal (47 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₅, S₃, C₆, C₆, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₁₅, C₄², C₂²⋊C₄, C₂²⋊C₄, C₂×D₄, C₂×Q₈, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, Dic₆, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂²×S₃, C₂²×C₆, D₁₅, C₃₀, C₃₀, C₄.₄D₄, Dic₁₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂²×D₅, C₂²×C₁₀, C₄×Dic₃, D₆⋊C₄, C₆.D₄, C₃×C₂²⋊C₄, C₂×Dic₆, C₂×C₃⋊D₄, Dic₁₅, Dic₁₅, C₆₀, D₃₀, C₂×C₃₀, C₂×C₃₀, C₄×Dic₅, D₁₀⋊C₄, C₂³.D₅, C₅×C₂²⋊C₄, C₂×Dic₁₀, C₂×C₅⋊D₄, C₂³.₁₁D₆, Dic₃₀, C₂×Dic₁₅, C₁₅⋊₇D₄, C₂×C₆₀, C₂²×D₁₅, C₂²×C₃₀, Dic₅.₅D₄, C₄×Dic₁₅, D₃₀⋊₃C₄, C₃₀.₃₈D₄, C₁₅×C₂²⋊C₄, C₂×Dic₃₀, C₂×C₁₅⋊₇D₄, C₂³.₁₁D₃₀
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, C₂×D₄, C₄○D₄, D₁₀, C₂²×S₃, D₁₅, C₄.₄D₄, C₂²×D₅, C₄○D₁₂, S₃×D₄, D₄⋊₂S₃, D₃₀, C₄○D₂₀, D₄×D₅, D₄⋊₂D₅, C₂³.₁₁D₆, C₂²×D₁₅, Dic₅.₅D₄, D₆₀⋊₁₁C₂, D₄×D₁₅, D₄⋊₂D₁₅, C₂³.₁₁D₃₀

Smallest permutation representation of C₂³.₁₁D₃₀
►On 240 points

Generators in S₂₄₀

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

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

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

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

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

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

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

84 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	4E	4F	4G	4H	5A	5B	6A	6B	6C	6D	6E	10A	···	10F	10G	10H	10I	10J	12A	12B	12C	12D	15A	15B	15C	15D	20A	···	20H	30A	···	30L	30M	···	30T	60A	···	60P
order	1	2	2	2	2	2	3	4	4	4	4	4	4	4	4	5	5	6	6	6	6	6	10	···	10	10	10	10	10	12	12	12	12	15	15	15	15	20	···	20	30	···	30	30	···	30	60	···	60
size	1	1	1	1	4	60	2	2	2	4	30	30	30	30	60	2	2	2	2	2	4	4	2	···	2	4	4	4	4	4	4	4	4	2	2	2	2	4	···	4	2	···	2	4	···	4	4	···	4

84 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+		+	+			+	-	+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₅	D₆	D₆	C₄○D₄	D₁₀	D₁₀	D₁₅	C₄○D₁₂	D₃₀	D₃₀	C₄○D₂₀	D₆₀⋊₁₁C₂	S₃×D₄	D₄⋊₂S₃	D₄×D₅	D₄⋊₂D₅	D₄×D₁₅	D₄⋊₂D₁₅
kernel	C₂³.₁₁D₃₀	C₄×Dic₁₅	D₃₀⋊₃C₄	C₃₀.₃₈D₄	C₁₅×C₂²⋊C₄	C₂×Dic₃₀	C₂×C₁₅⋊₇D₄	C₅×C₂²⋊C₄	Dic₁₅	C₃×C₂²⋊C₄	C₂×C₂₀	C₂²×C₁₀	C₃₀	C₂×C₁₂	C₂²×C₆	C₂²⋊C₄	C₁₀	C₂×C₄	C₂³	C₆	C₂	C₁₀	C₁₀	C₆	C₆	C₂	C₂
# reps	1	1	2	1	1	1	1	1	2	2	2	1	4	4	2	4	4	8	4	8	16	1	1	2	2	4	4

Matrix representation of C₂³.₁₁D₃₀ ►in GL₆(𝔽₆₁)

47	44	0	0	0	0
33	14	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	20	1

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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	60	0
0	0	0	0	0	60

4	11	0	0	0	0
54	11	0	0	0	0
0	0	0	60	0	0
0	0	1	1	0	0
0	0	0	0	60	6
0	0	0	0	20	1

58	9	0	0	0	0
26	3	0	0	0	0
0	0	0	60	0	0
0	0	60	0	0	0
0	0	0	0	1	55
0	0	0	0	0	60

G:=sub<GL(6,GF(61))| [47,33,0,0,0,0,44,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,20,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[4,54,0,0,0,0,11,11,0,0,0,0,0,0,0,1,0,0,0,0,60,1,0,0,0,0,0,0,60,20,0,0,0,0,6,1],[58,26,0,0,0,0,9,3,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,55,60] >;

C₂³.₁₁D₃₀ in GAP, Magma, Sage, TeX

C_2^3._{11}D_{30}

% in TeX

G:=Group("C2^3.11D30");

// GroupNames label

G:=SmallGroup(480,850);

// by ID

G=gap.SmallGroup(480,850);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,590,219,2693,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^30=b,e^2=c*b=b*c,e*a*e^-1=a*b=b*a,d*a*d^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^29>;

// generators/relations

G = C23.11D30 order 480 = 25·3·5

6th non-split extension by C23 of D30 acting via D30/C15=C22

G = C₂³.₁₁D₃₀ order 480 = 2⁵·3·5

6^th non-split extension by C₂³ of D₃₀ acting via D₃₀/C₁₅=C₂²