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G = C30.C24order 480 = 25·3·5

8th non-split extension by C30 of C24 acting via C24/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.8C24, D12.38D10, D30.2C23, C1532- 1+4, C60.161C23, Dic10.41D6, Dic6.40D10, D60.47C22, Dic15.6C23, Dic30.50C22, C4○D126D5, D15⋊Q812C2, C3⋊D4.3D10, C5⋊D12.C22, C6.8(C23×D5), C15⋊Q8.3C22, (C6×Dic10)⋊3C2, (C4×S3).14D10, (C2×C20).166D6, C10.8(S3×C23), D12⋊D511C2, D6011C26C2, D60⋊C211C2, D6.3(C22×D5), (S3×Dic10)⋊12C2, (C2×Dic10)⋊13S3, Dic3.D101C2, (C2×C12).165D10, (S3×C10).3C23, (C2×C60).35C22, (C2×Dic5).69D6, C51(Q8.15D6), C157D4.4C22, C31(D4.10D10), (S3×C20).29C22, C20.126(C22×S3), (C2×C30).227C23, (C4×D15).35C22, (C5×D12).44C22, C12.126(C22×D5), D30.C2.2C22, Dic5.5(C22×S3), Dic3.5(C22×D5), (C5×Dic3).5C23, (C3×Dic5).5C23, (S3×Dic5).2C22, (C5×Dic6).47C22, (C3×Dic10).42C22, (C6×Dic5).128C22, C4.133(C2×S3×D5), (C5×C4○D12)⋊2C2, C22.8(C2×S3×D5), (C2×C4).66(S3×D5), C2.12(C22×S3×D5), (C2×C10).9(C22×S3), (C5×C3⋊D4).3C22, (C2×C6).237(C22×D5), SmallGroup(480,1080)

Series: Derived Chief Lower central Upper central

C1C30 — C30.C24
C1C5C15C30C3×Dic5S3×Dic5Dic3.D10 — C30.C24
C15C30 — C30.C24
C1C2C2×C4

Generators and relations for C30.C24
 G = < a,b,c,d,e | a30=c2=d2=1, b2=e2=a15, bab-1=a19, cac=a11, ad=da, ae=ea, bc=cb, bd=db, ebe-1=a15b, dcd=a15c, ce=ec, de=ed >

Subgroups: 1340 in 292 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×8], C22, C22 [×4], C5, S3 [×4], C6, C6, C2×C4, C2×C4 [×14], D4 [×10], Q8 [×10], D5 [×2], C10, C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×4], D6 [×2], D6 [×2], C2×C6, C15, C2×Q8 [×5], C4○D4 [×10], Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], C2×C10, C2×C10 [×2], Dic6, Dic6 [×5], C4×S3 [×2], C4×S3 [×10], D12, D12 [×5], C3⋊D4 [×2], C3⋊D4 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×4], C5×S3 [×2], D15 [×2], C30, C30, 2- 1+4, Dic10 [×4], Dic10 [×5], C4×D5 [×6], D20, C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×6], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C4○D12, C4○D12 [×5], S3×Q8 [×4], Q83S3 [×4], C6×Q8, C5×Dic3 [×2], C3×Dic5 [×4], Dic15 [×2], C60 [×2], S3×C10 [×2], D30 [×2], C2×C30, C2×Dic10, C2×Dic10 [×2], C4○D20 [×3], D42D5 [×6], Q8×D5 [×2], C5×C4○D4, Q8.15D6, S3×Dic5 [×4], D30.C2 [×4], C5⋊D12 [×4], C15⋊Q8 [×4], C3×Dic10 [×4], C6×Dic5 [×2], C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], Dic30, C4×D15 [×2], D60, C157D4 [×2], C2×C60, D4.10D10, S3×Dic10 [×2], D12⋊D5 [×2], D60⋊C2 [×2], D15⋊Q8 [×2], Dic3.D10 [×4], C6×Dic10, C5×C4○D12, D6011C2, C30.C24
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2- 1+4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, Q8.15D6, C2×S3×D5 [×3], D4.10D10, C22×S3×D5, C30.C24

Smallest permutation representation of C30.C24
On 240 points
Generators in S240
(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)
(1 141 16 126)(2 130 17 145)(3 149 18 134)(4 138 19 123)(5 127 20 142)(6 146 21 131)(7 135 22 150)(8 124 23 139)(9 143 24 128)(10 132 25 147)(11 121 26 136)(12 140 27 125)(13 129 28 144)(14 148 29 133)(15 137 30 122)(31 172 46 157)(32 161 47 176)(33 180 48 165)(34 169 49 154)(35 158 50 173)(36 177 51 162)(37 166 52 151)(38 155 53 170)(39 174 54 159)(40 163 55 178)(41 152 56 167)(42 171 57 156)(43 160 58 175)(44 179 59 164)(45 168 60 153)(61 183 76 198)(62 202 77 187)(63 191 78 206)(64 210 79 195)(65 199 80 184)(66 188 81 203)(67 207 82 192)(68 196 83 181)(69 185 84 200)(70 204 85 189)(71 193 86 208)(72 182 87 197)(73 201 88 186)(74 190 89 205)(75 209 90 194)(91 230 106 215)(92 219 107 234)(93 238 108 223)(94 227 109 212)(95 216 110 231)(96 235 111 220)(97 224 112 239)(98 213 113 228)(99 232 114 217)(100 221 115 236)(101 240 116 225)(102 229 117 214)(103 218 118 233)(104 237 119 222)(105 226 120 211)
(1 83)(2 64)(3 75)(4 86)(5 67)(6 78)(7 89)(8 70)(9 81)(10 62)(11 73)(12 84)(13 65)(14 76)(15 87)(16 68)(17 79)(18 90)(19 71)(20 82)(21 63)(22 74)(23 85)(24 66)(25 77)(26 88)(27 69)(28 80)(29 61)(30 72)(31 119)(32 100)(33 111)(34 92)(35 103)(36 114)(37 95)(38 106)(39 117)(40 98)(41 109)(42 120)(43 101)(44 112)(45 93)(46 104)(47 115)(48 96)(49 107)(50 118)(51 99)(52 110)(53 91)(54 102)(55 113)(56 94)(57 105)(58 116)(59 97)(60 108)(121 201)(122 182)(123 193)(124 204)(125 185)(126 196)(127 207)(128 188)(129 199)(130 210)(131 191)(132 202)(133 183)(134 194)(135 205)(136 186)(137 197)(138 208)(139 189)(140 200)(141 181)(142 192)(143 203)(144 184)(145 195)(146 206)(147 187)(148 198)(149 209)(150 190)(151 231)(152 212)(153 223)(154 234)(155 215)(156 226)(157 237)(158 218)(159 229)(160 240)(161 221)(162 232)(163 213)(164 224)(165 235)(166 216)(167 227)(168 238)(169 219)(170 230)(171 211)(172 222)(173 233)(174 214)(175 225)(176 236)(177 217)(178 228)(179 239)(180 220)
(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(181 196)(182 197)(183 198)(184 199)(185 200)(186 201)(187 202)(188 203)(189 204)(190 205)(191 206)(192 207)(193 208)(194 209)(195 210)(211 226)(212 227)(213 228)(214 229)(215 230)(216 231)(217 232)(218 233)(219 234)(220 235)(221 236)(222 237)(223 238)(224 239)(225 240)
(1 57 16 42)(2 58 17 43)(3 59 18 44)(4 60 19 45)(5 31 20 46)(6 32 21 47)(7 33 22 48)(8 34 23 49)(9 35 24 50)(10 36 25 51)(11 37 26 52)(12 38 27 53)(13 39 28 54)(14 40 29 55)(15 41 30 56)(61 113 76 98)(62 114 77 99)(63 115 78 100)(64 116 79 101)(65 117 80 102)(66 118 81 103)(67 119 82 104)(68 120 83 105)(69 91 84 106)(70 92 85 107)(71 93 86 108)(72 94 87 109)(73 95 88 110)(74 96 89 111)(75 97 90 112)(121 151 136 166)(122 152 137 167)(123 153 138 168)(124 154 139 169)(125 155 140 170)(126 156 141 171)(127 157 142 172)(128 158 143 173)(129 159 144 174)(130 160 145 175)(131 161 146 176)(132 162 147 177)(133 163 148 178)(134 164 149 179)(135 165 150 180)(181 211 196 226)(182 212 197 227)(183 213 198 228)(184 214 199 229)(185 215 200 230)(186 216 201 231)(187 217 202 232)(188 218 203 233)(189 219 204 234)(190 220 205 235)(191 221 206 236)(192 222 207 237)(193 223 208 238)(194 224 209 239)(195 225 210 240)

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

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

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

63 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C10A10B10C10D10E10F10G10H12A12B12C12D12E12F15A15B20A20B20C20D20E20F20G20H20I20J30A···30F60A···60H
order12222223444444444455666101010101010101012121212121215152020202020202020202030···3060···60
size1126630302226610101010303022222224412121212442020202044222244121212124···44···4

63 irreducible representations

dim11111111122222222224444444
type+++++++++++++++++++-+++-
imageC1C2C2C2C2C2C2C2C2S3D5D6D6D6D10D10D10D10D102- 1+4S3×D5Q8.15D6C2×S3×D5C2×S3×D5D4.10D10C30.C24
kernelC30.C24S3×Dic10D12⋊D5D60⋊C2D15⋊Q8Dic3.D10C6×Dic10C5×C4○D12D6011C2C2×Dic10C4○D12Dic10C2×Dic5C2×C20Dic6C4×S3D12C3⋊D4C2×C12C15C2×C4C5C4C22C3C1
# reps12222411112421242421224248

Matrix representation of C30.C24 in GL4(𝔽61) generated by

01400
475300
00048
001351
,
11000
465000
00110
004650
,
0010
0001
1000
0100
,
1000
0100
00600
00060
,
255700
43600
002557
00436
G:=sub<GL(4,GF(61))| [0,47,0,0,14,53,0,0,0,0,0,13,0,0,48,51],[11,46,0,0,0,50,0,0,0,0,11,46,0,0,0,50],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[25,4,0,0,57,36,0,0,0,0,25,4,0,0,57,36] >;

C30.C24 in GAP, Magma, Sage, TeX

C_{30}.C_2^4
% in TeX

G:=Group("C30.C2^4");
// GroupNames label

G:=SmallGroup(480,1080);
// by ID

G=gap.SmallGroup(480,1080);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,219,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^30=c^2=d^2=1,b^2=e^2=a^15,b*a*b^-1=a^19,c*a*c=a^11,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=a^15*b,d*c*d=a^15*c,c*e=e*c,d*e=e*d>;
// generators/relations

׿
×
𝔽