C2^3.8D30 - GroupNames

G = C₂³.₈D₃₀ order 480 = 2⁵·3·5

3^rd non-split extension by C₂³ of D₃₀ acting via D₃₀/C₁₅=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₈D₃₀, C₆₀⋊₅C₄⋊₉C₂, (C₂×C₄).₂₆D₃₀, (C₂×C₁₂).₃₂D₁₀, (C₂×C₂₀).₂₀₆D₆, C₂²⋊C₄.₂D₁₅, C₃₀.₄Q₈⋊₅C₂, (C₄×Dic₁₅)⋊₁₆C₂, C₆.₉₅(C₄○D₂₀), (C₂×C₆₀).₁₇C₂², (C₂²×C₆).₅₄D₁₀, (C₂²×C₁₀).₆₉D₆, C₅⋊₇(C₂³.₈D₆), C₁₅⋊₁₈(C₄²⋊₂C₂), C₃₀.₂₁₅(C₄○D₄), C₁₀.₉₅(C₄○D₁₂), C₆.₉₀(D₄⋊₂D₅), C₂.₇(D₄⋊₂D₁₅), (C₂×C₃₀).₂₇₆C₂³, C₃₀.₃₈D₄.₃C₂, C₃⋊₇(C₂³.D₁₀), C₁₀.₉₀(D₄⋊₂S₃), C₂.₉(D₆₀⋊₁₁C₂), (C₂²×C₃₀).₁₀C₂², (C₂×Dic₁₅).₆C₂², C₂².₄₀(C₂²×D₁₅), (C₅×C₂²⋊C₄).₂S₃, (C₃×C₂²⋊C₄).₂D₅, (C₁₅×C₂²⋊C₄).₂C₂, (C₂×C₆).₂₇₂(C₂²×D₅), (C₂×C₁₀).₂₇₁(C₂²×S₃), SmallGroup(480,844)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₃₀ — C₂³.₈D₃₀

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₂×C₃₀ — C₂×Dic₁₅ — C₄×Dic₁₅ — C₂³.₈D₃₀

Lower central

C₁₅ — C₂×C₃₀ — C₂³.₈D₃₀

Upper central

C₁ — C₂² — C₂²⋊C₄

Generators and relations for C₂³.₈D₃₀
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=d²⁹ >

Subgroups: 548 in 120 conjugacy classes, 47 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₅, C₆, C₆, C₂×C₄, C₂×C₄, C₂³, C₁₀, C₁₀, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₁₅, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×Dic₃, C₂×C₁₂, C₂²×C₆, C₃₀, C₃₀, C₄²⋊₂C₂, C₂×Dic₅, C₂×C₂₀, C₂²×C₁₀, C₄×Dic₃, Dic₃⋊C₄, C₄⋊Dic₃, C₆.D₄, C₃×C₂²⋊C₄, Dic₁₅, C₆₀, C₂×C₃₀, C₂×C₃₀, C₄×Dic₅, C₁₀.D₄, C₄⋊Dic₅, C₂³.D₅, C₅×C₂²⋊C₄, C₂³.₈D₆, C₂×Dic₁₅, C₂×C₆₀, C₂²×C₃₀, C₂³.D₁₀, C₄×Dic₁₅, C₃₀.₄Q₈, C₆₀⋊₅C₄, C₃₀.₃₈D₄, C₁₅×C₂²⋊C₄, C₂³.₈D₃₀
Quotients: C₁, C₂, C₂², S₃, C₂³, D₅, D₆, C₄○D₄, D₁₀, C₂²×S₃, D₁₅, C₄²⋊₂C₂, C₂²×D₅, C₄○D₁₂, D₄⋊₂S₃, D₃₀, C₄○D₂₀, D₄⋊₂D₅, C₂³.₈D₆, C₂²×D₁₅, C₂³.D₁₀, D₆₀⋊₁₁C₂, D₄⋊₂D₁₅, C₂³.₈D₃₀

Smallest permutation representation of C₂³.₈D₃₀
►On 240 points

Generators in S₂₄₀

(2 147)(4 149)(6 151)(8 153)(10 155)(12 157)(14 159)(16 161)(18 163)(20 165)(22 167)(24 169)(26 171)(28 173)(30 175)(32 177)(34 179)(36 121)(38 123)(40 125)(42 127)(44 129)(46 131)(48 133)(50 135)(52 137)(54 139)(56 141)(58 143)(60 145)(61 91)(62 193)(63 93)(64 195)(65 95)(66 197)(67 97)(68 199)(69 99)(70 201)(71 101)(72 203)(73 103)(74 205)(75 105)(76 207)(77 107)(78 209)(79 109)(80 211)(81 111)(82 213)(83 113)(84 215)(85 115)(86 217)(87 117)(88 219)(89 119)(90 221)(92 223)(94 225)(96 227)(98 229)(100 231)(102 233)(104 235)(106 237)(108 239)(110 181)(112 183)(114 185)(116 187)(118 189)(120 191)(182 212)(184 214)(186 216)(188 218)(190 220)(192 222)(194 224)(196 226)(198 228)(200 230)(202 232)(204 234)(206 236)(208 238)(210 240)

(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)

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

(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)

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

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

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

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

84 conjugacy classes

class	1	2A	2B	2C	2D	3	4A	4B	4C	4D	4E	4F	4G	4H	4I	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	3	4	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	2	2	2	4	30	30	30	30	60	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	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+		+	+	+		+	+			-	-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₅	D₆	D₆	C₄○D₄	D₁₀	D₁₀	D₁₅	C₄○D₁₂	D₃₀	D₃₀	C₄○D₂₀	D₆₀⋊₁₁C₂	D₄⋊₂S₃	D₄⋊₂D₅	D₄⋊₂D₁₅
kernel	C₂³.₈D₃₀	C₄×Dic₁₅	C₃₀.₄Q₈	C₆₀⋊₅C₄	C₃₀.₃₈D₄	C₁₅×C₂²⋊C₄	C₅×C₂²⋊C₄	C₃×C₂²⋊C₄	C₂×C₂₀	C₂²×C₁₀	C₃₀	C₂×C₁₂	C₂²×C₆	C₂²⋊C₄	C₁₀	C₂×C₄	C₂³	C₆	C₂	C₁₀	C₆	C₂
# reps	1	1	2	1	2	1	1	2	2	1	6	4	2	4	4	8	4	8	16	2	4	8

Matrix representation of C₂³.₈D₃₀ ►in GL₆(𝔽₆₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	1	60	0	0
0	0	0	0	1	0
0	0	0	0	41	60

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	1	0
0	0	0	0	0	1

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

46	0	0	0	0	0
21	4	0	0	0	0
0	0	50	0	0	0
0	0	0	50	0	0
0	0	0	0	42	53
0	0	0	0	45	19

4	53	0	0	0	0
40	57	0	0	0	0
0	0	50	22	0	0
0	0	0	11	0	0
0	0	0	0	50	0
0	0	0	0	0	50

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,60,0,0,0,0,0,0,1,41,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[46,21,0,0,0,0,0,4,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,42,45,0,0,0,0,53,19],[4,40,0,0,0,0,53,57,0,0,0,0,0,0,50,0,0,0,0,0,22,11,0,0,0,0,0,0,50,0,0,0,0,0,0,50] >;

C₂³.₈D₃₀ in GAP, Magma, Sage, TeX

C_2^3._8D_{30}

% in TeX

G:=Group("C2^3.8D30");

// GroupNames label

G:=SmallGroup(480,844);

// by ID

G=gap.SmallGroup(480,844);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,590,219,2693,18822]);

// 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=d^29>;

// generators/relations

G = C23.8D30 order 480 = 25·3·5

3rd non-split extension by C23 of D30 acting via D30/C15=C22

G = C₂³.₈D₃₀ order 480 = 2⁵·3·5

3^rd non-split extension by C₂³ of D₃₀ acting via D₃₀/C₁₅=C₂²