C2^3.22D30 - GroupNames

G = C₂³.₂₂D₃₀ order 480 = 2⁵·3·5

8^th non-split extension by C₂³ of D₃₀ acting via D₃₀/D₁₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₂₂D₃₀, (C₂×C₃₀).₇D₄, (C₂×D₄).₅D₁₅, (C₆×D₄).₁₃D₅, (C₂×C₄).₁₇D₃₀, (D₄×C₁₀).₁₃S₃, (D₄×C₃₀).₂₄C₂, C₃₀.₃₈₁(C₂×D₄), (C₂×C₂₀).₂₅₀D₆, (C₂×C₁₂).₂₄₉D₁₀, C₃₀.₃₈D₄⋊₈C₂, C₃₀.₄Q₈⋊₃₆C₂, (C₂²×C₆).₆₃D₁₀, (C₂²×C₁₀).₇₈D₆, C₃₀.₂₂₃(C₄○D₄), (C₂×C₆₀).₄₃₃C₂², (C₂×C₃₀).₃₀₆C₂³, (C₂²×Dic₁₅)⋊₅C₂, C₅⋊₅(C₂³.₂₃D₆), C₂².₄(C₁₅⋊₇D₄), C₂.₁₅(D₄⋊₂D₁₅), C₆.₁₀₂(D₄⋊₂D₅), C₁₅⋊₃₇(C₂².D₄), C₃⋊₅(C₂³.₁₈D₁₀), (C₂²×C₃₀).₁₉C₂², C₁₀.₁₀₂(D₄⋊₂S₃), C₂².₅₇(C₂²×D₁₅), (C₂×Dic₁₅).₁₆C₂², C₆.₁₀₆(C₂×C₅⋊D₄), C₂.₁₁(C₂×C₁₅⋊₇D₄), (C₂×C₆).₁₉(C₅⋊D₄), C₁₀.₁₀₆(C₂×C₃⋊D₄), (C₂×C₁₀).₁₈(C₃⋊D₄), (C₂×C₆).₃₀₂(C₂²×D₅), (C₂×C₁₀).₃₀₁(C₂²×S₃), SmallGroup(480,900)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₃₀ — C₂³.₂₂D₃₀

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₂×C₃₀ — C₂×Dic₁₅ — C₂²×Dic₁₅ — C₂³.₂₂D₃₀

Lower central

C₁₅ — C₂×C₃₀ — C₂³.₂₂D₃₀

Upper central

C₁ — C₂² — C₂×D₄

Generators and relations for C₂³.₂₂D₃₀
G = < a,b,c,d,e | a²=b²=c²=d³⁰=1, e²=cb=bc, ab=ba, dad^-1=eae^-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede^-1=bd^-1 >

Subgroups: 692 in 156 conjugacy classes, 55 normal (31 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², C₅, C₆, C₆, C₆, C₂×C₄, C₂×C₄, D₄, C₂³, C₁₀, C₁₀, C₁₀, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₁₅, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×Dic₃, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₃₀, C₃₀, C₃₀, C₂².D₄, C₂×Dic₅, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, Dic₃⋊C₄, C₆.D₄, C₂²×Dic₃, C₆×D₄, Dic₁₅, C₆₀, C₂×C₃₀, C₂×C₃₀, C₂×C₃₀, C₁₀.D₄, C₂³.D₅, C₂²×Dic₅, D₄×C₁₀, C₂³.₂₃D₆, C₂×Dic₁₅, C₂×Dic₁₅, C₂×C₆₀, D₄×C₁₅, C₂²×C₃₀, C₂³.₁₈D₁₀, C₃₀.₄Q₈, C₃₀.₃₈D₄, C₃₀.₃₈D₄, C₂²×Dic₁₅, D₄×C₃₀, C₂³.₂₂D₃₀
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, C₂×D₄, C₄○D₄, D₁₀, C₃⋊D₄, C₂²×S₃, D₁₅, C₂².D₄, C₅⋊D₄, C₂²×D₅, D₄⋊₂S₃, C₂×C₃⋊D₄, D₃₀, D₄⋊₂D₅, C₂×C₅⋊D₄, C₂³.₂₃D₆, C₁₅⋊₇D₄, C₂²×D₁₅, C₂³.₁₈D₁₀, D₄⋊₂D₁₅, C₂×C₁₅⋊₇D₄, C₂³.₂₂D₃₀

Smallest permutation representation of C₂³.₂₂D₃₀
►On 240 points

Generators in S₂₄₀

(2 145)(4 147)(6 149)(8 121)(10 123)(12 125)(14 127)(16 129)(18 131)(20 133)(22 135)(24 137)(26 139)(28 141)(30 143)(31 107)(33 109)(35 111)(37 113)(39 115)(41 117)(43 119)(45 91)(47 93)(49 95)(51 97)(53 99)(55 101)(57 103)(59 105)(61 195)(63 197)(65 199)(67 201)(69 203)(71 205)(73 207)(75 209)(77 181)(79 183)(81 185)(83 187)(85 189)(87 191)(89 193)(151 231)(153 233)(155 235)(157 237)(159 239)(161 211)(163 213)(165 215)(167 217)(169 219)(171 221)(173 223)(175 225)(177 227)(179 229)

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

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

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

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

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

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

84 conjugacy classes

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

84 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+		+	+		+		+	+		-	-	-
image	C₁	C₂	C₂	C₂	C₂	S₃	D₄	D₅	D₆	D₆	C₄○D₄	D₁₀	D₁₀	C₃⋊D₄	D₁₅	C₅⋊D₄	D₃₀	D₃₀	C₁₅⋊₇D₄	D₄⋊₂S₃	D₄⋊₂D₅	D₄⋊₂D₁₅
kernel	C₂³.₂₂D₃₀	C₃₀.₄Q₈	C₃₀.₃₈D₄	C₂²×Dic₁₅	D₄×C₃₀	D₄×C₁₀	C₂×C₃₀	C₆×D₄	C₂×C₂₀	C₂²×C₁₀	C₃₀	C₂×C₁₂	C₂²×C₆	C₂×C₁₀	C₂×D₄	C₂×C₆	C₂×C₄	C₂³	C₂²	C₁₀	C₆	C₂
# reps	1	2	3	1	1	1	2	2	1	2	4	2	4	4	4	8	4	8	16	2	4	8

Matrix representation of C₂³.₂₂D₃₀ ►in GL₆(𝔽₆₁)

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	1	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

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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	47	9	0	0
0	0	22	25	0	0
0	0	0	0	0	1
0	0	0	0	1	0

18	52	0	0	0	0
9	43	0	0	0	0
0	0	49	59	0	0
0	0	41	12	0	0
0	0	0	0	0	50
0	0	0	0	11	0

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60],[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],[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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,47,22,0,0,0,0,9,25,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[18,9,0,0,0,0,52,43,0,0,0,0,0,0,49,41,0,0,0,0,59,12,0,0,0,0,0,0,0,11,0,0,0,0,50,0] >;

C₂³.₂₂D₃₀ in GAP, Magma, Sage, TeX

C_2^3._{22}D_{30}

% in TeX

G:=Group("C2^3.22D30");

// GroupNames label

G:=SmallGroup(480,900);

// by ID

G=gap.SmallGroup(480,900);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,219,2693,18822]);

// Polycyclic

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

// generators/relations

G = C23.22D30 order 480 = 25·3·5

8th non-split extension by C23 of D30 acting via D30/D15=C2

G = C₂³.₂₂D₃₀ order 480 = 2⁵·3·5

8^th non-split extension by C₂³ of D₃₀ acting via D₃₀/D₁₅=C₂