direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Q8.10D10, C30.78C24, C15⋊92- (1+4), C60.213C23, (Q8×D5)⋊7C6, C4○D20⋊6C6, (C6×Q8)⋊12D5, (Q8×C30)⋊14C2, Q8⋊2D5⋊7C6, (Q8×C10)⋊11C6, Q8.15(C6×D5), D20.13(C2×C6), (C3×Q8).42D10, C6.78(C23×D5), (C2×C12).253D10, C5⋊1(C3×2- (1+4)), C10.10(C23×C6), C20.24(C22×C6), (C6×D5).56C23, D10.5(C22×C6), (C2×C60).308C22, (C2×C30).386C23, Dic10.13(C2×C6), (D5×C12).80C22, (C3×D20).52C22, C12.213(C22×D5), (Q8×C15).47C22, Dic5.6(C22×C6), (C3×Dic5).58C23, (C3×Dic10).55C22, (C3×Q8×D5)⋊11C2, C4.24(D5×C2×C6), (C2×Q8)⋊7(C3×D5), C22.7(D5×C2×C6), (C4×D5).5(C2×C6), (C2×C4).19(C6×D5), C5⋊D4.2(C2×C6), (C3×C4○D20)⋊16C2, C2.11(D5×C22×C6), (C2×C20).45(C2×C6), (C5×Q8).18(C2×C6), (C3×Q8⋊2D5)⋊11C2, (C3×C5⋊D4).5C22, (C2×C10).68(C22×C6), (C2×C6).260(C22×D5), SmallGroup(480,1144)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 784 in 292 conjugacy classes, 170 normal (18 characteristic)
C1, C2, C2 [×5], C3, C4 [×6], C4 [×4], C22, C22 [×4], C5, C6, C6 [×5], C2×C4 [×3], C2×C4 [×12], D4 [×10], Q8 [×4], Q8 [×6], D5 [×4], C10, C10, C12 [×6], C12 [×4], C2×C6, C2×C6 [×4], C15, C2×Q8, C2×Q8 [×4], C4○D4 [×10], Dic5 [×4], C20 [×6], D10 [×4], C2×C10, C2×C12 [×3], C2×C12 [×12], C3×D4 [×10], C3×Q8 [×4], C3×Q8 [×6], C3×D5 [×4], C30, C30, 2- (1+4), Dic10 [×6], C4×D5 [×12], D20 [×6], C5⋊D4 [×4], C2×C20 [×3], C5×Q8 [×4], C6×Q8, C6×Q8 [×4], C3×C4○D4 [×10], C3×Dic5 [×4], C60 [×6], C6×D5 [×4], C2×C30, C4○D20 [×6], Q8×D5 [×4], Q8⋊2D5 [×4], Q8×C10, C3×2- (1+4), C3×Dic10 [×6], D5×C12 [×12], C3×D20 [×6], C3×C5⋊D4 [×4], C2×C60 [×3], Q8×C15 [×4], Q8.10D10, C3×C4○D20 [×6], C3×Q8×D5 [×4], C3×Q8⋊2D5 [×4], Q8×C30, C3×Q8.10D10
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], D5, C2×C6 [×35], C24, D10 [×7], C22×C6 [×15], C3×D5, 2- (1+4), C22×D5 [×7], C23×C6, C6×D5 [×7], C23×D5, C3×2- (1+4), D5×C2×C6 [×7], Q8.10D10, D5×C22×C6, C3×Q8.10D10
Generators and relations
G = < a,b,c,d,e | a3=b4=1, c2=d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ece-1=b2c, ede-1=d9 >
►On 240 points
Generators in S240
(1 87 190)(2 88 191)(3 89 192)(4 90 193)(5 91 194)(6 92 195)(7 93 196)(8 94 197)(9 95 198)(10 96 199)(11 97 200)(12 98 181)(13 99 182)(14 100 183)(15 81 184)(16 82 185)(17 83 186)(18 84 187)(19 85 188)(20 86 189)(21 70 111)(22 71 112)(23 72 113)(24 73 114)(25 74 115)(26 75 116)(27 76 117)(28 77 118)(29 78 119)(30 79 120)(31 80 101)(32 61 102)(33 62 103)(34 63 104)(35 64 105)(36 65 106)(37 66 107)(38 67 108)(39 68 109)(40 69 110)(41 175 127)(42 176 128)(43 177 129)(44 178 130)(45 179 131)(46 180 132)(47 161 133)(48 162 134)(49 163 135)(50 164 136)(51 165 137)(52 166 138)(53 167 139)(54 168 140)(55 169 121)(56 170 122)(57 171 123)(58 172 124)(59 173 125)(60 174 126)(141 211 230)(142 212 231)(143 213 232)(144 214 233)(145 215 234)(146 216 235)(147 217 236)(148 218 237)(149 219 238)(150 220 239)(151 201 240)(152 202 221)(153 203 222)(154 204 223)(155 205 224)(156 206 225)(157 207 226)(158 208 227)(159 209 228)(160 210 229)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)(81 86 91 96)(82 87 92 97)(83 88 93 98)(84 89 94 99)(85 90 95 100)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)(121 136 131 126)(122 137 132 127)(123 138 133 128)(124 139 134 129)(125 140 135 130)(141 156 151 146)(142 157 152 147)(143 158 153 148)(144 159 154 149)(145 160 155 150)(161 176 171 166)(162 177 172 167)(163 178 173 168)(164 179 174 169)(165 180 175 170)(181 186 191 196)(182 187 192 197)(183 188 193 198)(184 189 194 199)(185 190 195 200)(201 216 211 206)(202 217 212 207)(203 218 213 208)(204 219 214 209)(205 220 215 210)(221 236 231 226)(222 237 232 227)(223 238 233 228)(224 239 234 229)(225 240 235 230)
(1 69 11 79)(2 80 12 70)(3 71 13 61)(4 62 14 72)(5 73 15 63)(6 64 16 74)(7 75 17 65)(8 66 18 76)(9 77 19 67)(10 68 20 78)(21 191 31 181)(22 182 32 192)(23 193 33 183)(24 184 34 194)(25 195 35 185)(26 186 36 196)(27 197 37 187)(28 188 38 198)(29 199 39 189)(30 190 40 200)(41 239 51 229)(42 230 52 240)(43 221 53 231)(44 232 54 222)(45 223 55 233)(46 234 56 224)(47 225 57 235)(48 236 58 226)(49 227 59 237)(50 238 60 228)(81 104 91 114)(82 115 92 105)(83 106 93 116)(84 117 94 107)(85 108 95 118)(86 119 96 109)(87 110 97 120)(88 101 98 111)(89 112 99 102)(90 103 100 113)(121 214 131 204)(122 205 132 215)(123 216 133 206)(124 207 134 217)(125 218 135 208)(126 209 136 219)(127 220 137 210)(128 211 138 201)(129 202 139 212)(130 213 140 203)(141 166 151 176)(142 177 152 167)(143 168 153 178)(144 179 154 169)(145 170 155 180)(146 161 156 171)(147 172 157 162)(148 163 158 173)(149 174 159 164)(150 165 160 175)
(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)
(1 157 11 147)(2 146 12 156)(3 155 13 145)(4 144 14 154)(5 153 15 143)(6 142 16 152)(7 151 17 141)(8 160 18 150)(9 149 19 159)(10 158 20 148)(21 47 31 57)(22 56 32 46)(23 45 33 55)(24 54 34 44)(25 43 35 53)(26 52 36 42)(27 41 37 51)(28 50 38 60)(29 59 39 49)(30 48 40 58)(61 180 71 170)(62 169 72 179)(63 178 73 168)(64 167 74 177)(65 176 75 166)(66 165 76 175)(67 174 77 164)(68 163 78 173)(69 172 79 162)(70 161 80 171)(81 213 91 203)(82 202 92 212)(83 211 93 201)(84 220 94 210)(85 209 95 219)(86 218 96 208)(87 207 97 217)(88 216 98 206)(89 205 99 215)(90 214 100 204)(101 123 111 133)(102 132 112 122)(103 121 113 131)(104 130 114 140)(105 139 115 129)(106 128 116 138)(107 137 117 127)(108 126 118 136)(109 135 119 125)(110 124 120 134)(181 225 191 235)(182 234 192 224)(183 223 193 233)(184 232 194 222)(185 221 195 231)(186 230 196 240)(187 239 197 229)(188 228 198 238)(189 237 199 227)(190 226 200 236)
G:=sub<Sym(240)| (1,87,190)(2,88,191)(3,89,192)(4,90,193)(5,91,194)(6,92,195)(7,93,196)(8,94,197)(9,95,198)(10,96,199)(11,97,200)(12,98,181)(13,99,182)(14,100,183)(15,81,184)(16,82,185)(17,83,186)(18,84,187)(19,85,188)(20,86,189)(21,70,111)(22,71,112)(23,72,113)(24,73,114)(25,74,115)(26,75,116)(27,76,117)(28,77,118)(29,78,119)(30,79,120)(31,80,101)(32,61,102)(33,62,103)(34,63,104)(35,64,105)(36,65,106)(37,66,107)(38,67,108)(39,68,109)(40,69,110)(41,175,127)(42,176,128)(43,177,129)(44,178,130)(45,179,131)(46,180,132)(47,161,133)(48,162,134)(49,163,135)(50,164,136)(51,165,137)(52,166,138)(53,167,139)(54,168,140)(55,169,121)(56,170,122)(57,171,123)(58,172,124)(59,173,125)(60,174,126)(141,211,230)(142,212,231)(143,213,232)(144,214,233)(145,215,234)(146,216,235)(147,217,236)(148,218,237)(149,219,238)(150,220,239)(151,201,240)(152,202,221)(153,203,222)(154,204,223)(155,205,224)(156,206,225)(157,207,226)(158,208,227)(159,209,228)(160,210,229), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,26,31,36)(22,27,32,37)(23,28,33,38)(24,29,34,39)(25,30,35,40)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(61,66,71,76)(62,67,72,77)(63,68,73,78)(64,69,74,79)(65,70,75,80)(81,86,91,96)(82,87,92,97)(83,88,93,98)(84,89,94,99)(85,90,95,100)(101,106,111,116)(102,107,112,117)(103,108,113,118)(104,109,114,119)(105,110,115,120)(121,136,131,126)(122,137,132,127)(123,138,133,128)(124,139,134,129)(125,140,135,130)(141,156,151,146)(142,157,152,147)(143,158,153,148)(144,159,154,149)(145,160,155,150)(161,176,171,166)(162,177,172,167)(163,178,173,168)(164,179,174,169)(165,180,175,170)(181,186,191,196)(182,187,192,197)(183,188,193,198)(184,189,194,199)(185,190,195,200)(201,216,211,206)(202,217,212,207)(203,218,213,208)(204,219,214,209)(205,220,215,210)(221,236,231,226)(222,237,232,227)(223,238,233,228)(224,239,234,229)(225,240,235,230), (1,69,11,79)(2,80,12,70)(3,71,13,61)(4,62,14,72)(5,73,15,63)(6,64,16,74)(7,75,17,65)(8,66,18,76)(9,77,19,67)(10,68,20,78)(21,191,31,181)(22,182,32,192)(23,193,33,183)(24,184,34,194)(25,195,35,185)(26,186,36,196)(27,197,37,187)(28,188,38,198)(29,199,39,189)(30,190,40,200)(41,239,51,229)(42,230,52,240)(43,221,53,231)(44,232,54,222)(45,223,55,233)(46,234,56,224)(47,225,57,235)(48,236,58,226)(49,227,59,237)(50,238,60,228)(81,104,91,114)(82,115,92,105)(83,106,93,116)(84,117,94,107)(85,108,95,118)(86,119,96,109)(87,110,97,120)(88,101,98,111)(89,112,99,102)(90,103,100,113)(121,214,131,204)(122,205,132,215)(123,216,133,206)(124,207,134,217)(125,218,135,208)(126,209,136,219)(127,220,137,210)(128,211,138,201)(129,202,139,212)(130,213,140,203)(141,166,151,176)(142,177,152,167)(143,168,153,178)(144,179,154,169)(145,170,155,180)(146,161,156,171)(147,172,157,162)(148,163,158,173)(149,174,159,164)(150,165,160,175), (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), (1,157,11,147)(2,146,12,156)(3,155,13,145)(4,144,14,154)(5,153,15,143)(6,142,16,152)(7,151,17,141)(8,160,18,150)(9,149,19,159)(10,158,20,148)(21,47,31,57)(22,56,32,46)(23,45,33,55)(24,54,34,44)(25,43,35,53)(26,52,36,42)(27,41,37,51)(28,50,38,60)(29,59,39,49)(30,48,40,58)(61,180,71,170)(62,169,72,179)(63,178,73,168)(64,167,74,177)(65,176,75,166)(66,165,76,175)(67,174,77,164)(68,163,78,173)(69,172,79,162)(70,161,80,171)(81,213,91,203)(82,202,92,212)(83,211,93,201)(84,220,94,210)(85,209,95,219)(86,218,96,208)(87,207,97,217)(88,216,98,206)(89,205,99,215)(90,214,100,204)(101,123,111,133)(102,132,112,122)(103,121,113,131)(104,130,114,140)(105,139,115,129)(106,128,116,138)(107,137,117,127)(108,126,118,136)(109,135,119,125)(110,124,120,134)(181,225,191,235)(182,234,192,224)(183,223,193,233)(184,232,194,222)(185,221,195,231)(186,230,196,240)(187,239,197,229)(188,228,198,238)(189,237,199,227)(190,226,200,236)>;
G:=Group( (1,87,190)(2,88,191)(3,89,192)(4,90,193)(5,91,194)(6,92,195)(7,93,196)(8,94,197)(9,95,198)(10,96,199)(11,97,200)(12,98,181)(13,99,182)(14,100,183)(15,81,184)(16,82,185)(17,83,186)(18,84,187)(19,85,188)(20,86,189)(21,70,111)(22,71,112)(23,72,113)(24,73,114)(25,74,115)(26,75,116)(27,76,117)(28,77,118)(29,78,119)(30,79,120)(31,80,101)(32,61,102)(33,62,103)(34,63,104)(35,64,105)(36,65,106)(37,66,107)(38,67,108)(39,68,109)(40,69,110)(41,175,127)(42,176,128)(43,177,129)(44,178,130)(45,179,131)(46,180,132)(47,161,133)(48,162,134)(49,163,135)(50,164,136)(51,165,137)(52,166,138)(53,167,139)(54,168,140)(55,169,121)(56,170,122)(57,171,123)(58,172,124)(59,173,125)(60,174,126)(141,211,230)(142,212,231)(143,213,232)(144,214,233)(145,215,234)(146,216,235)(147,217,236)(148,218,237)(149,219,238)(150,220,239)(151,201,240)(152,202,221)(153,203,222)(154,204,223)(155,205,224)(156,206,225)(157,207,226)(158,208,227)(159,209,228)(160,210,229), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,26,31,36)(22,27,32,37)(23,28,33,38)(24,29,34,39)(25,30,35,40)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(61,66,71,76)(62,67,72,77)(63,68,73,78)(64,69,74,79)(65,70,75,80)(81,86,91,96)(82,87,92,97)(83,88,93,98)(84,89,94,99)(85,90,95,100)(101,106,111,116)(102,107,112,117)(103,108,113,118)(104,109,114,119)(105,110,115,120)(121,136,131,126)(122,137,132,127)(123,138,133,128)(124,139,134,129)(125,140,135,130)(141,156,151,146)(142,157,152,147)(143,158,153,148)(144,159,154,149)(145,160,155,150)(161,176,171,166)(162,177,172,167)(163,178,173,168)(164,179,174,169)(165,180,175,170)(181,186,191,196)(182,187,192,197)(183,188,193,198)(184,189,194,199)(185,190,195,200)(201,216,211,206)(202,217,212,207)(203,218,213,208)(204,219,214,209)(205,220,215,210)(221,236,231,226)(222,237,232,227)(223,238,233,228)(224,239,234,229)(225,240,235,230), (1,69,11,79)(2,80,12,70)(3,71,13,61)(4,62,14,72)(5,73,15,63)(6,64,16,74)(7,75,17,65)(8,66,18,76)(9,77,19,67)(10,68,20,78)(21,191,31,181)(22,182,32,192)(23,193,33,183)(24,184,34,194)(25,195,35,185)(26,186,36,196)(27,197,37,187)(28,188,38,198)(29,199,39,189)(30,190,40,200)(41,239,51,229)(42,230,52,240)(43,221,53,231)(44,232,54,222)(45,223,55,233)(46,234,56,224)(47,225,57,235)(48,236,58,226)(49,227,59,237)(50,238,60,228)(81,104,91,114)(82,115,92,105)(83,106,93,116)(84,117,94,107)(85,108,95,118)(86,119,96,109)(87,110,97,120)(88,101,98,111)(89,112,99,102)(90,103,100,113)(121,214,131,204)(122,205,132,215)(123,216,133,206)(124,207,134,217)(125,218,135,208)(126,209,136,219)(127,220,137,210)(128,211,138,201)(129,202,139,212)(130,213,140,203)(141,166,151,176)(142,177,152,167)(143,168,153,178)(144,179,154,169)(145,170,155,180)(146,161,156,171)(147,172,157,162)(148,163,158,173)(149,174,159,164)(150,165,160,175), (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), (1,157,11,147)(2,146,12,156)(3,155,13,145)(4,144,14,154)(5,153,15,143)(6,142,16,152)(7,151,17,141)(8,160,18,150)(9,149,19,159)(10,158,20,148)(21,47,31,57)(22,56,32,46)(23,45,33,55)(24,54,34,44)(25,43,35,53)(26,52,36,42)(27,41,37,51)(28,50,38,60)(29,59,39,49)(30,48,40,58)(61,180,71,170)(62,169,72,179)(63,178,73,168)(64,167,74,177)(65,176,75,166)(66,165,76,175)(67,174,77,164)(68,163,78,173)(69,172,79,162)(70,161,80,171)(81,213,91,203)(82,202,92,212)(83,211,93,201)(84,220,94,210)(85,209,95,219)(86,218,96,208)(87,207,97,217)(88,216,98,206)(89,205,99,215)(90,214,100,204)(101,123,111,133)(102,132,112,122)(103,121,113,131)(104,130,114,140)(105,139,115,129)(106,128,116,138)(107,137,117,127)(108,126,118,136)(109,135,119,125)(110,124,120,134)(181,225,191,235)(182,234,192,224)(183,223,193,233)(184,232,194,222)(185,221,195,231)(186,230,196,240)(187,239,197,229)(188,228,198,238)(189,237,199,227)(190,226,200,236) );
G=PermutationGroup([(1,87,190),(2,88,191),(3,89,192),(4,90,193),(5,91,194),(6,92,195),(7,93,196),(8,94,197),(9,95,198),(10,96,199),(11,97,200),(12,98,181),(13,99,182),(14,100,183),(15,81,184),(16,82,185),(17,83,186),(18,84,187),(19,85,188),(20,86,189),(21,70,111),(22,71,112),(23,72,113),(24,73,114),(25,74,115),(26,75,116),(27,76,117),(28,77,118),(29,78,119),(30,79,120),(31,80,101),(32,61,102),(33,62,103),(34,63,104),(35,64,105),(36,65,106),(37,66,107),(38,67,108),(39,68,109),(40,69,110),(41,175,127),(42,176,128),(43,177,129),(44,178,130),(45,179,131),(46,180,132),(47,161,133),(48,162,134),(49,163,135),(50,164,136),(51,165,137),(52,166,138),(53,167,139),(54,168,140),(55,169,121),(56,170,122),(57,171,123),(58,172,124),(59,173,125),(60,174,126),(141,211,230),(142,212,231),(143,213,232),(144,214,233),(145,215,234),(146,216,235),(147,217,236),(148,218,237),(149,219,238),(150,220,239),(151,201,240),(152,202,221),(153,203,222),(154,204,223),(155,205,224),(156,206,225),(157,207,226),(158,208,227),(159,209,228),(160,210,229)], [(1,6,11,16),(2,7,12,17),(3,8,13,18),(4,9,14,19),(5,10,15,20),(21,26,31,36),(22,27,32,37),(23,28,33,38),(24,29,34,39),(25,30,35,40),(41,56,51,46),(42,57,52,47),(43,58,53,48),(44,59,54,49),(45,60,55,50),(61,66,71,76),(62,67,72,77),(63,68,73,78),(64,69,74,79),(65,70,75,80),(81,86,91,96),(82,87,92,97),(83,88,93,98),(84,89,94,99),(85,90,95,100),(101,106,111,116),(102,107,112,117),(103,108,113,118),(104,109,114,119),(105,110,115,120),(121,136,131,126),(122,137,132,127),(123,138,133,128),(124,139,134,129),(125,140,135,130),(141,156,151,146),(142,157,152,147),(143,158,153,148),(144,159,154,149),(145,160,155,150),(161,176,171,166),(162,177,172,167),(163,178,173,168),(164,179,174,169),(165,180,175,170),(181,186,191,196),(182,187,192,197),(183,188,193,198),(184,189,194,199),(185,190,195,200),(201,216,211,206),(202,217,212,207),(203,218,213,208),(204,219,214,209),(205,220,215,210),(221,236,231,226),(222,237,232,227),(223,238,233,228),(224,239,234,229),(225,240,235,230)], [(1,69,11,79),(2,80,12,70),(3,71,13,61),(4,62,14,72),(5,73,15,63),(6,64,16,74),(7,75,17,65),(8,66,18,76),(9,77,19,67),(10,68,20,78),(21,191,31,181),(22,182,32,192),(23,193,33,183),(24,184,34,194),(25,195,35,185),(26,186,36,196),(27,197,37,187),(28,188,38,198),(29,199,39,189),(30,190,40,200),(41,239,51,229),(42,230,52,240),(43,221,53,231),(44,232,54,222),(45,223,55,233),(46,234,56,224),(47,225,57,235),(48,236,58,226),(49,227,59,237),(50,238,60,228),(81,104,91,114),(82,115,92,105),(83,106,93,116),(84,117,94,107),(85,108,95,118),(86,119,96,109),(87,110,97,120),(88,101,98,111),(89,112,99,102),(90,103,100,113),(121,214,131,204),(122,205,132,215),(123,216,133,206),(124,207,134,217),(125,218,135,208),(126,209,136,219),(127,220,137,210),(128,211,138,201),(129,202,139,212),(130,213,140,203),(141,166,151,176),(142,177,152,167),(143,168,153,178),(144,179,154,169),(145,170,155,180),(146,161,156,171),(147,172,157,162),(148,163,158,173),(149,174,159,164),(150,165,160,175)], [(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)], [(1,157,11,147),(2,146,12,156),(3,155,13,145),(4,144,14,154),(5,153,15,143),(6,142,16,152),(7,151,17,141),(8,160,18,150),(9,149,19,159),(10,158,20,148),(21,47,31,57),(22,56,32,46),(23,45,33,55),(24,54,34,44),(25,43,35,53),(26,52,36,42),(27,41,37,51),(28,50,38,60),(29,59,39,49),(30,48,40,58),(61,180,71,170),(62,169,72,179),(63,178,73,168),(64,167,74,177),(65,176,75,166),(66,165,76,175),(67,174,77,164),(68,163,78,173),(69,172,79,162),(70,161,80,171),(81,213,91,203),(82,202,92,212),(83,211,93,201),(84,220,94,210),(85,209,95,219),(86,218,96,208),(87,207,97,217),(88,216,98,206),(89,205,99,215),(90,214,100,204),(101,123,111,133),(102,132,112,122),(103,121,113,131),(104,130,114,140),(105,139,115,129),(106,128,116,138),(107,137,117,127),(108,126,118,136),(109,135,119,125),(110,124,120,134),(181,225,191,235),(182,234,192,224),(183,223,193,233),(184,232,194,222),(185,221,195,231),(186,230,196,240),(187,239,197,229),(188,228,198,238),(189,237,199,227),(190,226,200,236)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 60 | 15 | 0 | 0 |
| 8 | 1 | 0 | 0 |
| 37 | 37 | 0 | 60 |
| 15 | 43 | 1 | 0 |
| 8 | 11 | 0 | 0 |
| 44 | 53 | 0 | 0 |
| 18 | 36 | 25 | 17 |
| 19 | 60 | 17 | 36 |
| 41 | 56 | 0 | 0 |
| 38 | 20 | 0 | 0 |
| 49 | 49 | 0 | 58 |
| 2 | 22 | 3 | 0 |
| 36 | 36 | 0 | 6 |
| 4 | 20 | 24 | 37 |
| 25 | 20 | 19 | 5 |
| 14 | 30 | 39 | 47 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[60,8,37,15,15,1,37,43,0,0,0,1,0,0,60,0],[8,44,18,19,11,53,36,60,0,0,25,17,0,0,17,36],[41,38,49,2,56,20,49,22,0,0,0,3,0,0,58,0],[36,4,25,14,36,20,20,30,0,24,19,39,6,37,5,47] >;
111 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 10A | ··· | 10F | 12A | ··· | 12L | 12M | ··· | 12T | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 10 | 10 | 10 | 10 | 1 | 1 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
111 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D5 | D10 | D10 | C3×D5 | C6×D5 | C6×D5 | 2- (1+4) | C3×2- (1+4) | Q8.10D10 | C3×Q8.10D10 |
| kernel | C3×Q8.10D10 | C3×C4○D20 | C3×Q8×D5 | C3×Q8⋊2D5 | Q8×C30 | Q8.10D10 | C4○D20 | Q8×D5 | Q8⋊2D5 | Q8×C10 | C6×Q8 | C2×C12 | C3×Q8 | C2×Q8 | C2×C4 | Q8 | C15 | C5 | C3 | C1 |
| # reps | 1 | 6 | 4 | 4 | 1 | 2 | 12 | 8 | 8 | 2 | 2 | 6 | 8 | 4 | 12 | 16 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_3\times Q_8._{10}D_{10} % in TeX
G:=Group("C3xQ8.10D10"); // GroupNames label
G:=SmallGroup(480,1144);
// by ID
G=gap.SmallGroup(480,1144);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,344,555,268,1571,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=1,c^2=d^10=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=e*c*e^-1=b^2*c,e*d*e^-1=d^9>;
// generators/relations
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