direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4○D4×C30, C60.303C23, C30.101C24, (C2×D4)⋊7C30, D4⋊3(C2×C30), Q8⋊4(C2×C30), (C2×Q8)⋊8C30, (C6×D4)⋊16C10, (D4×C30)⋊34C2, (D4×C10)⋊16C6, (C22×C4)⋊8C30, (C6×Q8)⋊13C10, (Q8×C30)⋊27C2, (Q8×C10)⋊17C6, (C2×C60)⋊54C22, (C22×C20)⋊17C6, (C22×C60)⋊27C2, C4.8(C22×C30), C2.3(C23×C30), (C22×C12)⋊13C10, (D4×C15)⋊43C22, C20.51(C22×C6), C23.13(C2×C30), C10.18(C23×C6), C6.18(C23×C10), (Q8×C15)⋊38C22, (C2×C30).262C23, C12.55(C22×C10), C22.1(C22×C30), (C22×C30).134C22, (C2×C4)⋊5(C2×C30), (C2×C20)⋊16(C2×C6), (C5×D4)⋊12(C2×C6), (C5×Q8)⋊13(C2×C6), (C2×C12)⋊16(C2×C10), (C3×D4)⋊12(C2×C10), (C3×Q8)⋊11(C2×C10), (C2×C10).6(C22×C6), (C2×C6).6(C22×C10), (C22×C10).38(C2×C6), (C22×C6).30(C2×C10), SmallGroup(480,1183)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 376 in 328 conjugacy classes, 280 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×8], C22, C22 [×6], C22 [×6], C5, C6, C6 [×2], C6 [×6], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C10, C10 [×2], C10 [×6], C12 [×8], C2×C6, C2×C6 [×6], C2×C6 [×6], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×8], C2×C10, C2×C10 [×6], C2×C10 [×6], C2×C12, C2×C12 [×15], C3×D4 [×12], C3×Q8 [×4], C22×C6 [×3], C30, C30 [×2], C30 [×6], C2×C4○D4, C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], C22×C12 [×3], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], C60 [×8], C2×C30, C2×C30 [×6], C2×C30 [×6], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C6×C4○D4, C2×C60, C2×C60 [×15], D4×C15 [×12], Q8×C15 [×4], C22×C30 [×3], C10×C4○D4, C22×C60 [×3], D4×C30 [×3], Q8×C30, C15×C4○D4 [×8], C4○D4×C30
Quotients:
C1, C2 [×15], C3, C22 [×35], C5, C6 [×15], C23 [×15], C10 [×15], C2×C6 [×35], C15, C4○D4 [×2], C24, C2×C10 [×35], C22×C6 [×15], C30 [×15], C2×C4○D4, C22×C10 [×15], C3×C4○D4 [×2], C23×C6, C2×C30 [×35], C5×C4○D4 [×2], C23×C10, C6×C4○D4, C22×C30 [×15], C10×C4○D4, C15×C4○D4 [×2], C23×C30, C4○D4×C30
Generators and relations
G = < a,b,c,d | a30=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
►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)
(1 115 154 130)(2 116 155 131)(3 117 156 132)(4 118 157 133)(5 119 158 134)(6 120 159 135)(7 91 160 136)(8 92 161 137)(9 93 162 138)(10 94 163 139)(11 95 164 140)(12 96 165 141)(13 97 166 142)(14 98 167 143)(15 99 168 144)(16 100 169 145)(17 101 170 146)(18 102 171 147)(19 103 172 148)(20 104 173 149)(21 105 174 150)(22 106 175 121)(23 107 176 122)(24 108 177 123)(25 109 178 124)(26 110 179 125)(27 111 180 126)(28 112 151 127)(29 113 152 128)(30 114 153 129)(31 78 220 197)(32 79 221 198)(33 80 222 199)(34 81 223 200)(35 82 224 201)(36 83 225 202)(37 84 226 203)(38 85 227 204)(39 86 228 205)(40 87 229 206)(41 88 230 207)(42 89 231 208)(43 90 232 209)(44 61 233 210)(45 62 234 181)(46 63 235 182)(47 64 236 183)(48 65 237 184)(49 66 238 185)(50 67 239 186)(51 68 240 187)(52 69 211 188)(53 70 212 189)(54 71 213 190)(55 72 214 191)(56 73 215 192)(57 74 216 193)(58 75 217 194)(59 76 218 195)(60 77 219 196)
(1 194 154 75)(2 195 155 76)(3 196 156 77)(4 197 157 78)(5 198 158 79)(6 199 159 80)(7 200 160 81)(8 201 161 82)(9 202 162 83)(10 203 163 84)(11 204 164 85)(12 205 165 86)(13 206 166 87)(14 207 167 88)(15 208 168 89)(16 209 169 90)(17 210 170 61)(18 181 171 62)(19 182 172 63)(20 183 173 64)(21 184 174 65)(22 185 175 66)(23 186 176 67)(24 187 177 68)(25 188 178 69)(26 189 179 70)(27 190 180 71)(28 191 151 72)(29 192 152 73)(30 193 153 74)(31 133 220 118)(32 134 221 119)(33 135 222 120)(34 136 223 91)(35 137 224 92)(36 138 225 93)(37 139 226 94)(38 140 227 95)(39 141 228 96)(40 142 229 97)(41 143 230 98)(42 144 231 99)(43 145 232 100)(44 146 233 101)(45 147 234 102)(46 148 235 103)(47 149 236 104)(48 150 237 105)(49 121 238 106)(50 122 239 107)(51 123 240 108)(52 124 211 109)(53 125 212 110)(54 126 213 111)(55 127 214 112)(56 128 215 113)(57 129 216 114)(58 130 217 115)(59 131 218 116)(60 132 219 117)
(1 169)(2 170)(3 171)(4 172)(5 173)(6 174)(7 175)(8 176)(9 177)(10 178)(11 179)(12 180)(13 151)(14 152)(15 153)(16 154)(17 155)(18 156)(19 157)(20 158)(21 159)(22 160)(23 161)(24 162)(25 163)(26 164)(27 165)(28 166)(29 167)(30 168)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(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)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)(101 131)(102 132)(103 133)(104 134)(105 135)(106 136)(107 137)(108 138)(109 139)(110 140)(111 141)(112 142)(113 143)(114 144)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)(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)
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), (1,115,154,130)(2,116,155,131)(3,117,156,132)(4,118,157,133)(5,119,158,134)(6,120,159,135)(7,91,160,136)(8,92,161,137)(9,93,162,138)(10,94,163,139)(11,95,164,140)(12,96,165,141)(13,97,166,142)(14,98,167,143)(15,99,168,144)(16,100,169,145)(17,101,170,146)(18,102,171,147)(19,103,172,148)(20,104,173,149)(21,105,174,150)(22,106,175,121)(23,107,176,122)(24,108,177,123)(25,109,178,124)(26,110,179,125)(27,111,180,126)(28,112,151,127)(29,113,152,128)(30,114,153,129)(31,78,220,197)(32,79,221,198)(33,80,222,199)(34,81,223,200)(35,82,224,201)(36,83,225,202)(37,84,226,203)(38,85,227,204)(39,86,228,205)(40,87,229,206)(41,88,230,207)(42,89,231,208)(43,90,232,209)(44,61,233,210)(45,62,234,181)(46,63,235,182)(47,64,236,183)(48,65,237,184)(49,66,238,185)(50,67,239,186)(51,68,240,187)(52,69,211,188)(53,70,212,189)(54,71,213,190)(55,72,214,191)(56,73,215,192)(57,74,216,193)(58,75,217,194)(59,76,218,195)(60,77,219,196), (1,194,154,75)(2,195,155,76)(3,196,156,77)(4,197,157,78)(5,198,158,79)(6,199,159,80)(7,200,160,81)(8,201,161,82)(9,202,162,83)(10,203,163,84)(11,204,164,85)(12,205,165,86)(13,206,166,87)(14,207,167,88)(15,208,168,89)(16,209,169,90)(17,210,170,61)(18,181,171,62)(19,182,172,63)(20,183,173,64)(21,184,174,65)(22,185,175,66)(23,186,176,67)(24,187,177,68)(25,188,178,69)(26,189,179,70)(27,190,180,71)(28,191,151,72)(29,192,152,73)(30,193,153,74)(31,133,220,118)(32,134,221,119)(33,135,222,120)(34,136,223,91)(35,137,224,92)(36,138,225,93)(37,139,226,94)(38,140,227,95)(39,141,228,96)(40,142,229,97)(41,143,230,98)(42,144,231,99)(43,145,232,100)(44,146,233,101)(45,147,234,102)(46,148,235,103)(47,149,236,104)(48,150,237,105)(49,121,238,106)(50,122,239,107)(51,123,240,108)(52,124,211,109)(53,125,212,110)(54,126,213,111)(55,127,214,112)(56,128,215,113)(57,129,216,114)(58,130,217,115)(59,131,218,116)(60,132,219,117), (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,151)(14,152)(15,153)(16,154)(17,155)(18,156)(19,157)(20,158)(21,159)(22,160)(23,161)(24,162)(25,163)(26,164)(27,165)(28,166)(29,167)(30,168)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(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)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130)(101,131)(102,132)(103,133)(104,134)(105,135)(106,136)(107,137)(108,138)(109,139)(110,140)(111,141)(112,142)(113,143)(114,144)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150)(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)>;
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), (1,115,154,130)(2,116,155,131)(3,117,156,132)(4,118,157,133)(5,119,158,134)(6,120,159,135)(7,91,160,136)(8,92,161,137)(9,93,162,138)(10,94,163,139)(11,95,164,140)(12,96,165,141)(13,97,166,142)(14,98,167,143)(15,99,168,144)(16,100,169,145)(17,101,170,146)(18,102,171,147)(19,103,172,148)(20,104,173,149)(21,105,174,150)(22,106,175,121)(23,107,176,122)(24,108,177,123)(25,109,178,124)(26,110,179,125)(27,111,180,126)(28,112,151,127)(29,113,152,128)(30,114,153,129)(31,78,220,197)(32,79,221,198)(33,80,222,199)(34,81,223,200)(35,82,224,201)(36,83,225,202)(37,84,226,203)(38,85,227,204)(39,86,228,205)(40,87,229,206)(41,88,230,207)(42,89,231,208)(43,90,232,209)(44,61,233,210)(45,62,234,181)(46,63,235,182)(47,64,236,183)(48,65,237,184)(49,66,238,185)(50,67,239,186)(51,68,240,187)(52,69,211,188)(53,70,212,189)(54,71,213,190)(55,72,214,191)(56,73,215,192)(57,74,216,193)(58,75,217,194)(59,76,218,195)(60,77,219,196), (1,194,154,75)(2,195,155,76)(3,196,156,77)(4,197,157,78)(5,198,158,79)(6,199,159,80)(7,200,160,81)(8,201,161,82)(9,202,162,83)(10,203,163,84)(11,204,164,85)(12,205,165,86)(13,206,166,87)(14,207,167,88)(15,208,168,89)(16,209,169,90)(17,210,170,61)(18,181,171,62)(19,182,172,63)(20,183,173,64)(21,184,174,65)(22,185,175,66)(23,186,176,67)(24,187,177,68)(25,188,178,69)(26,189,179,70)(27,190,180,71)(28,191,151,72)(29,192,152,73)(30,193,153,74)(31,133,220,118)(32,134,221,119)(33,135,222,120)(34,136,223,91)(35,137,224,92)(36,138,225,93)(37,139,226,94)(38,140,227,95)(39,141,228,96)(40,142,229,97)(41,143,230,98)(42,144,231,99)(43,145,232,100)(44,146,233,101)(45,147,234,102)(46,148,235,103)(47,149,236,104)(48,150,237,105)(49,121,238,106)(50,122,239,107)(51,123,240,108)(52,124,211,109)(53,125,212,110)(54,126,213,111)(55,127,214,112)(56,128,215,113)(57,129,216,114)(58,130,217,115)(59,131,218,116)(60,132,219,117), (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,151)(14,152)(15,153)(16,154)(17,155)(18,156)(19,157)(20,158)(21,159)(22,160)(23,161)(24,162)(25,163)(26,164)(27,165)(28,166)(29,167)(30,168)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(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)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130)(101,131)(102,132)(103,133)(104,134)(105,135)(106,136)(107,137)(108,138)(109,139)(110,140)(111,141)(112,142)(113,143)(114,144)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150)(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) );
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)], [(1,115,154,130),(2,116,155,131),(3,117,156,132),(4,118,157,133),(5,119,158,134),(6,120,159,135),(7,91,160,136),(8,92,161,137),(9,93,162,138),(10,94,163,139),(11,95,164,140),(12,96,165,141),(13,97,166,142),(14,98,167,143),(15,99,168,144),(16,100,169,145),(17,101,170,146),(18,102,171,147),(19,103,172,148),(20,104,173,149),(21,105,174,150),(22,106,175,121),(23,107,176,122),(24,108,177,123),(25,109,178,124),(26,110,179,125),(27,111,180,126),(28,112,151,127),(29,113,152,128),(30,114,153,129),(31,78,220,197),(32,79,221,198),(33,80,222,199),(34,81,223,200),(35,82,224,201),(36,83,225,202),(37,84,226,203),(38,85,227,204),(39,86,228,205),(40,87,229,206),(41,88,230,207),(42,89,231,208),(43,90,232,209),(44,61,233,210),(45,62,234,181),(46,63,235,182),(47,64,236,183),(48,65,237,184),(49,66,238,185),(50,67,239,186),(51,68,240,187),(52,69,211,188),(53,70,212,189),(54,71,213,190),(55,72,214,191),(56,73,215,192),(57,74,216,193),(58,75,217,194),(59,76,218,195),(60,77,219,196)], [(1,194,154,75),(2,195,155,76),(3,196,156,77),(4,197,157,78),(5,198,158,79),(6,199,159,80),(7,200,160,81),(8,201,161,82),(9,202,162,83),(10,203,163,84),(11,204,164,85),(12,205,165,86),(13,206,166,87),(14,207,167,88),(15,208,168,89),(16,209,169,90),(17,210,170,61),(18,181,171,62),(19,182,172,63),(20,183,173,64),(21,184,174,65),(22,185,175,66),(23,186,176,67),(24,187,177,68),(25,188,178,69),(26,189,179,70),(27,190,180,71),(28,191,151,72),(29,192,152,73),(30,193,153,74),(31,133,220,118),(32,134,221,119),(33,135,222,120),(34,136,223,91),(35,137,224,92),(36,138,225,93),(37,139,226,94),(38,140,227,95),(39,141,228,96),(40,142,229,97),(41,143,230,98),(42,144,231,99),(43,145,232,100),(44,146,233,101),(45,147,234,102),(46,148,235,103),(47,149,236,104),(48,150,237,105),(49,121,238,106),(50,122,239,107),(51,123,240,108),(52,124,211,109),(53,125,212,110),(54,126,213,111),(55,127,214,112),(56,128,215,113),(57,129,216,114),(58,130,217,115),(59,131,218,116),(60,132,219,117)], [(1,169),(2,170),(3,171),(4,172),(5,173),(6,174),(7,175),(8,176),(9,177),(10,178),(11,179),(12,180),(13,151),(14,152),(15,153),(16,154),(17,155),(18,156),(19,157),(20,158),(21,159),(22,160),(23,161),(24,162),(25,163),(26,164),(27,165),(28,166),(29,167),(30,168),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60),(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),(91,121),(92,122),(93,123),(94,124),(95,125),(96,126),(97,127),(98,128),(99,129),(100,130),(101,131),(102,132),(103,133),(104,134),(105,135),(106,136),(107,137),(108,138),(109,139),(110,140),(111,141),(112,142),(113,143),(114,144),(115,145),(116,146),(117,147),(118,148),(119,149),(120,150),(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)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 34 |
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 2 |
| 0 | 0 | 60 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 60 | 1 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,13,0,0,0,0,34,0,0,0,0,34],[1,0,0,0,0,60,0,0,0,0,50,0,0,0,0,50],[1,0,0,0,0,60,0,0,0,0,60,60,0,0,2,1],[1,0,0,0,0,60,0,0,0,0,60,60,0,0,0,1] >;
300 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | ··· | 6R | 10A | ··· | 10L | 10M | ··· | 10AJ | 12A | ··· | 12H | 12I | ··· | 12T | 15A | ··· | 15H | 20A | ··· | 20P | 20Q | ··· | 20AN | 30A | ··· | 30X | 30Y | ··· | 30BT | 60A | ··· | 60AF | 60AG | ··· | 60CB |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
300 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C6 | C10 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | C30 | C4○D4 | C3×C4○D4 | C5×C4○D4 | C15×C4○D4 |
| kernel | C4○D4×C30 | C22×C60 | D4×C30 | Q8×C30 | C15×C4○D4 | C10×C4○D4 | C6×C4○D4 | C22×C20 | D4×C10 | Q8×C10 | C5×C4○D4 | C22×C12 | C6×D4 | C6×Q8 | C3×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C30 | C10 | C6 | C2 |
| # reps | 1 | 3 | 3 | 1 | 8 | 2 | 4 | 6 | 6 | 2 | 16 | 12 | 12 | 4 | 32 | 8 | 24 | 24 | 8 | 64 | 4 | 8 | 16 | 32 |
In GAP, Magma, Sage, TeX
C_4\circ D_4\times C_{30} % in TeX
G:=Group("C4oD4xC30"); // GroupNames label
G:=SmallGroup(480,1183);
// by ID
G=gap.SmallGroup(480,1183);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-5,-2,3389,1276]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀