metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.15D30, C30.63C24, C60.87C23, C15⋊82- (1+4), D30.28C23, D60.42C22, Dic30.44C22, Dic15.30C23, (C6×Q8)⋊7D5, (C2×Q8)⋊7D15, (Q8×C30)⋊7C2, (Q8×C10)⋊11S3, (Q8×D15)⋊11C2, (C2×C4).23D30, (C5×Q8).53D6, (C2×C20).173D6, (C3×Q8).36D10, Q8⋊3D15⋊11C2, C6.63(C23×D5), (C2×C12).171D10, C10.63(S3×C23), (C2×C60).89C22, C5⋊4(Q8.15D6), D60⋊11C2⋊16C2, C4.24(C22×D15), C2.11(C23×D15), C15⋊7D4.5C22, (C2×C30).324C23, C20.137(C22×S3), C3⋊4(Q8.10D10), (C4×D15).28C22, C12.135(C22×D5), (Q8×C15).41C22, C22.7(C22×D15), (C2×C6).320(C22×D5), (C2×C10).320(C22×S3), SmallGroup(480,1174)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1492 in 292 conjugacy classes, 119 normal (17 characteristic)
C1, C2, C2 [×5], C3, C4 [×6], C4 [×4], C22, C22 [×4], C5, S3 [×4], C6, C6, C2×C4 [×3], C2×C4 [×12], D4 [×10], Q8 [×4], Q8 [×6], D5 [×4], C10, C10, Dic3 [×4], C12 [×6], D6 [×4], C2×C6, C15, C2×Q8, C2×Q8 [×4], C4○D4 [×10], Dic5 [×4], C20 [×6], D10 [×4], C2×C10, Dic6 [×6], C4×S3 [×12], D12 [×6], C3⋊D4 [×4], C2×C12 [×3], C3×Q8 [×4], D15 [×4], C30, C30, 2- (1+4), Dic10 [×6], C4×D5 [×12], D20 [×6], C5⋊D4 [×4], C2×C20 [×3], C5×Q8 [×4], C4○D12 [×6], S3×Q8 [×4], Q8⋊3S3 [×4], C6×Q8, Dic15 [×4], C60 [×6], D30 [×4], C2×C30, C4○D20 [×6], Q8×D5 [×4], Q8⋊2D5 [×4], Q8×C10, Q8.15D6, Dic30 [×6], C4×D15 [×12], D60 [×6], C15⋊7D4 [×4], C2×C60 [×3], Q8×C15 [×4], Q8.10D10, D60⋊11C2 [×6], Q8×D15 [×4], Q8⋊3D15 [×4], Q8×C30, Q8.15D30
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], D15, 2- (1+4), C22×D5 [×7], S3×C23, D30 [×7], C23×D5, Q8.15D6, C22×D15 [×7], Q8.10D10, C23×D15, Q8.15D30
Generators and relations
G = < a,b,c,d | a4=1, b2=c30=d2=a2, bab-1=cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c29 >
►On 240 points
Generators in S240
(1 66 31 96)(2 97 32 67)(3 68 33 98)(4 99 34 69)(5 70 35 100)(6 101 36 71)(7 72 37 102)(8 103 38 73)(9 74 39 104)(10 105 40 75)(11 76 41 106)(12 107 42 77)(13 78 43 108)(14 109 44 79)(15 80 45 110)(16 111 46 81)(17 82 47 112)(18 113 48 83)(19 84 49 114)(20 115 50 85)(21 86 51 116)(22 117 52 87)(23 88 53 118)(24 119 54 89)(25 90 55 120)(26 61 56 91)(27 92 57 62)(28 63 58 93)(29 94 59 64)(30 65 60 95)(121 195 151 225)(122 226 152 196)(123 197 153 227)(124 228 154 198)(125 199 155 229)(126 230 156 200)(127 201 157 231)(128 232 158 202)(129 203 159 233)(130 234 160 204)(131 205 161 235)(132 236 162 206)(133 207 163 237)(134 238 164 208)(135 209 165 239)(136 240 166 210)(137 211 167 181)(138 182 168 212)(139 213 169 183)(140 184 170 214)(141 215 171 185)(142 186 172 216)(143 217 173 187)(144 188 174 218)(145 219 175 189)(146 190 176 220)(147 221 177 191)(148 192 178 222)(149 223 179 193)(150 194 180 224)
(1 46 31 16)(2 47 32 17)(3 48 33 18)(4 49 34 19)(5 50 35 20)(6 51 36 21)(7 52 37 22)(8 53 38 23)(9 54 39 24)(10 55 40 25)(11 56 41 26)(12 57 42 27)(13 58 43 28)(14 59 44 29)(15 60 45 30)(61 106 91 76)(62 107 92 77)(63 108 93 78)(64 109 94 79)(65 110 95 80)(66 111 96 81)(67 112 97 82)(68 113 98 83)(69 114 99 84)(70 115 100 85)(71 116 101 86)(72 117 102 87)(73 118 103 88)(74 119 104 89)(75 120 105 90)(121 136 151 166)(122 137 152 167)(123 138 153 168)(124 139 154 169)(125 140 155 170)(126 141 156 171)(127 142 157 172)(128 143 158 173)(129 144 159 174)(130 145 160 175)(131 146 161 176)(132 147 162 177)(133 148 163 178)(134 149 164 179)(135 150 165 180)(181 196 211 226)(182 197 212 227)(183 198 213 228)(184 199 214 229)(185 200 215 230)(186 201 216 231)(187 202 217 232)(188 203 218 233)(189 204 219 234)(190 205 220 235)(191 206 221 236)(192 207 222 237)(193 208 223 238)(194 209 224 239)(195 210 225 240)
(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)
(1 178 31 148)(2 147 32 177)(3 176 33 146)(4 145 34 175)(5 174 35 144)(6 143 36 173)(7 172 37 142)(8 141 38 171)(9 170 39 140)(10 139 40 169)(11 168 41 138)(12 137 42 167)(13 166 43 136)(14 135 44 165)(15 164 45 134)(16 133 46 163)(17 162 47 132)(18 131 48 161)(19 160 49 130)(20 129 50 159)(21 158 51 128)(22 127 52 157)(23 156 53 126)(24 125 54 155)(25 154 55 124)(26 123 56 153)(27 152 57 122)(28 121 58 151)(29 150 59 180)(30 179 60 149)(61 227 91 197)(62 196 92 226)(63 225 93 195)(64 194 94 224)(65 223 95 193)(66 192 96 222)(67 221 97 191)(68 190 98 220)(69 219 99 189)(70 188 100 218)(71 217 101 187)(72 186 102 216)(73 215 103 185)(74 184 104 214)(75 213 105 183)(76 182 106 212)(77 211 107 181)(78 240 108 210)(79 209 109 239)(80 238 110 208)(81 207 111 237)(82 236 112 206)(83 205 113 235)(84 234 114 204)(85 203 115 233)(86 232 116 202)(87 201 117 231)(88 230 118 200)(89 199 119 229)(90 228 120 198)
G:=sub<Sym(240)| (1,66,31,96)(2,97,32,67)(3,68,33,98)(4,99,34,69)(5,70,35,100)(6,101,36,71)(7,72,37,102)(8,103,38,73)(9,74,39,104)(10,105,40,75)(11,76,41,106)(12,107,42,77)(13,78,43,108)(14,109,44,79)(15,80,45,110)(16,111,46,81)(17,82,47,112)(18,113,48,83)(19,84,49,114)(20,115,50,85)(21,86,51,116)(22,117,52,87)(23,88,53,118)(24,119,54,89)(25,90,55,120)(26,61,56,91)(27,92,57,62)(28,63,58,93)(29,94,59,64)(30,65,60,95)(121,195,151,225)(122,226,152,196)(123,197,153,227)(124,228,154,198)(125,199,155,229)(126,230,156,200)(127,201,157,231)(128,232,158,202)(129,203,159,233)(130,234,160,204)(131,205,161,235)(132,236,162,206)(133,207,163,237)(134,238,164,208)(135,209,165,239)(136,240,166,210)(137,211,167,181)(138,182,168,212)(139,213,169,183)(140,184,170,214)(141,215,171,185)(142,186,172,216)(143,217,173,187)(144,188,174,218)(145,219,175,189)(146,190,176,220)(147,221,177,191)(148,192,178,222)(149,223,179,193)(150,194,180,224), (1,46,31,16)(2,47,32,17)(3,48,33,18)(4,49,34,19)(5,50,35,20)(6,51,36,21)(7,52,37,22)(8,53,38,23)(9,54,39,24)(10,55,40,25)(11,56,41,26)(12,57,42,27)(13,58,43,28)(14,59,44,29)(15,60,45,30)(61,106,91,76)(62,107,92,77)(63,108,93,78)(64,109,94,79)(65,110,95,80)(66,111,96,81)(67,112,97,82)(68,113,98,83)(69,114,99,84)(70,115,100,85)(71,116,101,86)(72,117,102,87)(73,118,103,88)(74,119,104,89)(75,120,105,90)(121,136,151,166)(122,137,152,167)(123,138,153,168)(124,139,154,169)(125,140,155,170)(126,141,156,171)(127,142,157,172)(128,143,158,173)(129,144,159,174)(130,145,160,175)(131,146,161,176)(132,147,162,177)(133,148,163,178)(134,149,164,179)(135,150,165,180)(181,196,211,226)(182,197,212,227)(183,198,213,228)(184,199,214,229)(185,200,215,230)(186,201,216,231)(187,202,217,232)(188,203,218,233)(189,204,219,234)(190,205,220,235)(191,206,221,236)(192,207,222,237)(193,208,223,238)(194,209,224,239)(195,210,225,240), (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), (1,178,31,148)(2,147,32,177)(3,176,33,146)(4,145,34,175)(5,174,35,144)(6,143,36,173)(7,172,37,142)(8,141,38,171)(9,170,39,140)(10,139,40,169)(11,168,41,138)(12,137,42,167)(13,166,43,136)(14,135,44,165)(15,164,45,134)(16,133,46,163)(17,162,47,132)(18,131,48,161)(19,160,49,130)(20,129,50,159)(21,158,51,128)(22,127,52,157)(23,156,53,126)(24,125,54,155)(25,154,55,124)(26,123,56,153)(27,152,57,122)(28,121,58,151)(29,150,59,180)(30,179,60,149)(61,227,91,197)(62,196,92,226)(63,225,93,195)(64,194,94,224)(65,223,95,193)(66,192,96,222)(67,221,97,191)(68,190,98,220)(69,219,99,189)(70,188,100,218)(71,217,101,187)(72,186,102,216)(73,215,103,185)(74,184,104,214)(75,213,105,183)(76,182,106,212)(77,211,107,181)(78,240,108,210)(79,209,109,239)(80,238,110,208)(81,207,111,237)(82,236,112,206)(83,205,113,235)(84,234,114,204)(85,203,115,233)(86,232,116,202)(87,201,117,231)(88,230,118,200)(89,199,119,229)(90,228,120,198)>;
G:=Group( (1,66,31,96)(2,97,32,67)(3,68,33,98)(4,99,34,69)(5,70,35,100)(6,101,36,71)(7,72,37,102)(8,103,38,73)(9,74,39,104)(10,105,40,75)(11,76,41,106)(12,107,42,77)(13,78,43,108)(14,109,44,79)(15,80,45,110)(16,111,46,81)(17,82,47,112)(18,113,48,83)(19,84,49,114)(20,115,50,85)(21,86,51,116)(22,117,52,87)(23,88,53,118)(24,119,54,89)(25,90,55,120)(26,61,56,91)(27,92,57,62)(28,63,58,93)(29,94,59,64)(30,65,60,95)(121,195,151,225)(122,226,152,196)(123,197,153,227)(124,228,154,198)(125,199,155,229)(126,230,156,200)(127,201,157,231)(128,232,158,202)(129,203,159,233)(130,234,160,204)(131,205,161,235)(132,236,162,206)(133,207,163,237)(134,238,164,208)(135,209,165,239)(136,240,166,210)(137,211,167,181)(138,182,168,212)(139,213,169,183)(140,184,170,214)(141,215,171,185)(142,186,172,216)(143,217,173,187)(144,188,174,218)(145,219,175,189)(146,190,176,220)(147,221,177,191)(148,192,178,222)(149,223,179,193)(150,194,180,224), (1,46,31,16)(2,47,32,17)(3,48,33,18)(4,49,34,19)(5,50,35,20)(6,51,36,21)(7,52,37,22)(8,53,38,23)(9,54,39,24)(10,55,40,25)(11,56,41,26)(12,57,42,27)(13,58,43,28)(14,59,44,29)(15,60,45,30)(61,106,91,76)(62,107,92,77)(63,108,93,78)(64,109,94,79)(65,110,95,80)(66,111,96,81)(67,112,97,82)(68,113,98,83)(69,114,99,84)(70,115,100,85)(71,116,101,86)(72,117,102,87)(73,118,103,88)(74,119,104,89)(75,120,105,90)(121,136,151,166)(122,137,152,167)(123,138,153,168)(124,139,154,169)(125,140,155,170)(126,141,156,171)(127,142,157,172)(128,143,158,173)(129,144,159,174)(130,145,160,175)(131,146,161,176)(132,147,162,177)(133,148,163,178)(134,149,164,179)(135,150,165,180)(181,196,211,226)(182,197,212,227)(183,198,213,228)(184,199,214,229)(185,200,215,230)(186,201,216,231)(187,202,217,232)(188,203,218,233)(189,204,219,234)(190,205,220,235)(191,206,221,236)(192,207,222,237)(193,208,223,238)(194,209,224,239)(195,210,225,240), (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), (1,178,31,148)(2,147,32,177)(3,176,33,146)(4,145,34,175)(5,174,35,144)(6,143,36,173)(7,172,37,142)(8,141,38,171)(9,170,39,140)(10,139,40,169)(11,168,41,138)(12,137,42,167)(13,166,43,136)(14,135,44,165)(15,164,45,134)(16,133,46,163)(17,162,47,132)(18,131,48,161)(19,160,49,130)(20,129,50,159)(21,158,51,128)(22,127,52,157)(23,156,53,126)(24,125,54,155)(25,154,55,124)(26,123,56,153)(27,152,57,122)(28,121,58,151)(29,150,59,180)(30,179,60,149)(61,227,91,197)(62,196,92,226)(63,225,93,195)(64,194,94,224)(65,223,95,193)(66,192,96,222)(67,221,97,191)(68,190,98,220)(69,219,99,189)(70,188,100,218)(71,217,101,187)(72,186,102,216)(73,215,103,185)(74,184,104,214)(75,213,105,183)(76,182,106,212)(77,211,107,181)(78,240,108,210)(79,209,109,239)(80,238,110,208)(81,207,111,237)(82,236,112,206)(83,205,113,235)(84,234,114,204)(85,203,115,233)(86,232,116,202)(87,201,117,231)(88,230,118,200)(89,199,119,229)(90,228,120,198) );
G=PermutationGroup([(1,66,31,96),(2,97,32,67),(3,68,33,98),(4,99,34,69),(5,70,35,100),(6,101,36,71),(7,72,37,102),(8,103,38,73),(9,74,39,104),(10,105,40,75),(11,76,41,106),(12,107,42,77),(13,78,43,108),(14,109,44,79),(15,80,45,110),(16,111,46,81),(17,82,47,112),(18,113,48,83),(19,84,49,114),(20,115,50,85),(21,86,51,116),(22,117,52,87),(23,88,53,118),(24,119,54,89),(25,90,55,120),(26,61,56,91),(27,92,57,62),(28,63,58,93),(29,94,59,64),(30,65,60,95),(121,195,151,225),(122,226,152,196),(123,197,153,227),(124,228,154,198),(125,199,155,229),(126,230,156,200),(127,201,157,231),(128,232,158,202),(129,203,159,233),(130,234,160,204),(131,205,161,235),(132,236,162,206),(133,207,163,237),(134,238,164,208),(135,209,165,239),(136,240,166,210),(137,211,167,181),(138,182,168,212),(139,213,169,183),(140,184,170,214),(141,215,171,185),(142,186,172,216),(143,217,173,187),(144,188,174,218),(145,219,175,189),(146,190,176,220),(147,221,177,191),(148,192,178,222),(149,223,179,193),(150,194,180,224)], [(1,46,31,16),(2,47,32,17),(3,48,33,18),(4,49,34,19),(5,50,35,20),(6,51,36,21),(7,52,37,22),(8,53,38,23),(9,54,39,24),(10,55,40,25),(11,56,41,26),(12,57,42,27),(13,58,43,28),(14,59,44,29),(15,60,45,30),(61,106,91,76),(62,107,92,77),(63,108,93,78),(64,109,94,79),(65,110,95,80),(66,111,96,81),(67,112,97,82),(68,113,98,83),(69,114,99,84),(70,115,100,85),(71,116,101,86),(72,117,102,87),(73,118,103,88),(74,119,104,89),(75,120,105,90),(121,136,151,166),(122,137,152,167),(123,138,153,168),(124,139,154,169),(125,140,155,170),(126,141,156,171),(127,142,157,172),(128,143,158,173),(129,144,159,174),(130,145,160,175),(131,146,161,176),(132,147,162,177),(133,148,163,178),(134,149,164,179),(135,150,165,180),(181,196,211,226),(182,197,212,227),(183,198,213,228),(184,199,214,229),(185,200,215,230),(186,201,216,231),(187,202,217,232),(188,203,218,233),(189,204,219,234),(190,205,220,235),(191,206,221,236),(192,207,222,237),(193,208,223,238),(194,209,224,239),(195,210,225,240)], [(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)], [(1,178,31,148),(2,147,32,177),(3,176,33,146),(4,145,34,175),(5,174,35,144),(6,143,36,173),(7,172,37,142),(8,141,38,171),(9,170,39,140),(10,139,40,169),(11,168,41,138),(12,137,42,167),(13,166,43,136),(14,135,44,165),(15,164,45,134),(16,133,46,163),(17,162,47,132),(18,131,48,161),(19,160,49,130),(20,129,50,159),(21,158,51,128),(22,127,52,157),(23,156,53,126),(24,125,54,155),(25,154,55,124),(26,123,56,153),(27,152,57,122),(28,121,58,151),(29,150,59,180),(30,179,60,149),(61,227,91,197),(62,196,92,226),(63,225,93,195),(64,194,94,224),(65,223,95,193),(66,192,96,222),(67,221,97,191),(68,190,98,220),(69,219,99,189),(70,188,100,218),(71,217,101,187),(72,186,102,216),(73,215,103,185),(74,184,104,214),(75,213,105,183),(76,182,106,212),(77,211,107,181),(78,240,108,210),(79,209,109,239),(80,238,110,208),(81,207,111,237),(82,236,112,206),(83,205,113,235),(84,234,114,204),(85,203,115,233),(86,232,116,202),(87,201,117,231),(88,230,118,200),(89,199,119,229),(90,228,120,198)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 0 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 19 | 59 | 0 | 0 |
| 59 | 42 | 0 | 0 |
| 0 | 0 | 42 | 2 |
| 0 | 0 | 2 | 19 |
| 60 | 29 | 0 | 0 |
| 29 | 1 | 0 | 0 |
| 0 | 0 | 5 | 38 |
| 0 | 0 | 38 | 56 |
| 0 | 0 | 5 | 38 |
| 0 | 0 | 38 | 56 |
| 60 | 29 | 0 | 0 |
| 29 | 1 | 0 | 0 |
G:=sub<GL(4,GF(61))| [0,60,0,0,1,0,0,0,0,0,0,60,0,0,1,0],[19,59,0,0,59,42,0,0,0,0,42,2,0,0,2,19],[60,29,0,0,29,1,0,0,0,0,5,38,0,0,38,56],[0,0,60,29,0,0,29,1,5,38,0,0,38,56,0,0] >;
87 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | ··· | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
87 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D10 | D10 | D15 | D30 | D30 | 2- (1+4) | Q8.15D6 | Q8.10D10 | Q8.15D30 |
| kernel | Q8.15D30 | D60⋊11C2 | Q8×D15 | Q8⋊3D15 | Q8×C30 | Q8×C10 | C6×Q8 | C2×C20 | C5×Q8 | C2×C12 | C3×Q8 | C2×Q8 | C2×C4 | Q8 | C15 | C5 | C3 | C1 |
| # reps | 1 | 6 | 4 | 4 | 1 | 1 | 2 | 3 | 4 | 6 | 8 | 4 | 12 | 16 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
Q_8._{15}D_{30} % in TeX
G:=Group("Q8.15D30"); // GroupNames label
G:=SmallGroup(480,1174);
// by ID
G=gap.SmallGroup(480,1174);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=1,b^2=c^30=d^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^29>;
// generators/relations
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