metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.38D6, C30.4C24, D30.1C23, C15⋊12- (1+4), C60.159C23, Dic6.41D10, Dic10.40D6, D60.52C22, Dic15.3C23, Dic30.55C22, C4○D20⋊5S3, C5⋊1(Q8○D12), C5⋊D4.3D6, D15⋊Q8⋊11C2, (C4×D5).13D6, C3⋊D20.C22, C6.4(C23×D5), C15⋊Q8.1C22, (C2×Dic6)⋊13D5, (D5×Dic6)⋊11C2, (C10×Dic6)⋊3C2, (C2×C20).163D6, C10.4(S3×C23), D60⋊11C2⋊3C2, C12.28D10⋊11C2, D20⋊S3⋊11C2, (C6×D5).3C23, Dic5.D6⋊1C2, (C2×C12).161D10, (C2×C60).28C22, D10.3(C22×S3), C15⋊7D4.3C22, (C2×C30).223C23, C20.123(C22×S3), (C2×Dic3).68D10, C3⋊1(Q8.10D10), (D5×C12).28C22, (C4×D15).34C22, (C3×D20).44C22, C12.159(C22×D5), D30.C2.1C22, Dic3.3(C22×D5), (C3×Dic5).1C23, Dic5.1(C22×S3), (D5×Dic3).1C22, (C5×Dic3).3C23, (C5×Dic6).41C22, (C3×Dic10).47C22, (C10×Dic3).128C22, C4.84(C2×S3×D5), (C3×C4○D20)⋊2C2, C2.8(C22×S3×D5), (C2×C4).64(S3×D5), C22.13(C2×S3×D5), (C2×C6).8(C22×D5), (C3×C5⋊D4).2C22, (C2×C10).235(C22×S3), SmallGroup(480,1076)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1356 in 292 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×8], C22, C22 [×4], C5, S3 [×2], C6, C6 [×3], C2×C4, C2×C4 [×14], D4 [×10], Q8 [×10], D5 [×4], C10, C10, Dic3 [×4], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], C2×C6, C2×C6 [×2], C15, C2×Q8 [×5], C4○D4 [×10], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×4], D10 [×2], D10 [×2], C2×C10, Dic6 [×4], Dic6 [×5], C4×S3 [×6], D12, C2×Dic3 [×2], C2×Dic3 [×4], C3⋊D4 [×6], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C3×D5 [×2], D15 [×2], C30, C30, 2- (1+4), Dic10, Dic10 [×5], C4×D5 [×2], C4×D5 [×10], D20, D20 [×5], C5⋊D4 [×2], C5⋊D4 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C2×Dic6, C2×Dic6 [×2], C4○D12 [×3], D4⋊2S3 [×6], S3×Q8 [×2], C3×C4○D4, C5×Dic3 [×4], C3×Dic5 [×2], Dic15 [×2], C60 [×2], C6×D5 [×2], D30 [×2], C2×C30, C4○D20, C4○D20 [×5], Q8×D5 [×4], Q8⋊2D5 [×4], Q8×C10, Q8○D12, D5×Dic3 [×4], D30.C2 [×4], C3⋊D20 [×4], C15⋊Q8 [×4], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], C5×Dic6 [×4], C10×Dic3 [×2], Dic30, C4×D15 [×2], D60, C15⋊7D4 [×2], C2×C60, Q8.10D10, D5×Dic6 [×2], D20⋊S3 [×2], D15⋊Q8 [×2], C12.28D10 [×2], Dic5.D6 [×4], C3×C4○D20, C10×Dic6, D60⋊11C2, D20.38D6
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○D12, C2×S3×D5 [×3], Q8.10D10, C22×S3×D5, D20.38D6
Generators and relations
G = < a,b,c,d | a20=b2=1, c6=d2=a10, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=a10c5 >
►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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 60)(18 59)(19 58)(20 57)(21 158)(22 157)(23 156)(24 155)(25 154)(26 153)(27 152)(28 151)(29 150)(30 149)(31 148)(32 147)(33 146)(34 145)(35 144)(36 143)(37 142)(38 141)(39 160)(40 159)(61 179)(62 178)(63 177)(64 176)(65 175)(66 174)(67 173)(68 172)(69 171)(70 170)(71 169)(72 168)(73 167)(74 166)(75 165)(76 164)(77 163)(78 162)(79 161)(80 180)(81 102)(82 101)(83 120)(84 119)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(121 217)(122 216)(123 215)(124 214)(125 213)(126 212)(127 211)(128 210)(129 209)(130 208)(131 207)(132 206)(133 205)(134 204)(135 203)(136 202)(137 201)(138 220)(139 219)(140 218)(181 226)(182 225)(183 224)(184 223)(185 222)(186 221)(187 240)(188 239)(189 238)(190 237)(191 236)(192 235)(193 234)(194 233)(195 232)(196 231)(197 230)(198 229)(199 228)(200 227)
(1 94 151 16 89 146 11 84 141 6 99 156)(2 95 152 17 90 147 12 85 142 7 100 157)(3 96 153 18 91 148 13 86 143 8 81 158)(4 97 154 19 92 149 14 87 144 9 82 159)(5 98 155 20 93 150 15 88 145 10 83 160)(21 54 107 26 59 112 31 44 117 36 49 102)(22 55 108 27 60 113 32 45 118 37 50 103)(23 56 109 28 41 114 33 46 119 38 51 104)(24 57 110 29 42 115 34 47 120 39 52 105)(25 58 111 30 43 116 35 48 101 40 53 106)(61 197 126 66 182 131 71 187 136 76 192 121)(62 198 127 67 183 132 72 188 137 77 193 122)(63 199 128 68 184 133 73 189 138 78 194 123)(64 200 129 69 185 134 74 190 139 79 195 124)(65 181 130 70 186 135 75 191 140 80 196 125)(161 232 214 176 227 209 171 222 204 166 237 219)(162 233 215 177 228 210 172 223 205 167 238 220)(163 234 216 178 229 211 173 224 206 168 239 201)(164 235 217 179 230 212 174 225 207 169 240 202)(165 236 218 180 231 213 175 226 208 170 221 203)
(1 227 11 237)(2 238 12 228)(3 229 13 239)(4 240 14 230)(5 231 15 221)(6 222 16 232)(7 233 17 223)(8 224 18 234)(9 235 19 225)(10 226 20 236)(21 137 31 127)(22 128 32 138)(23 139 33 129)(24 130 34 140)(25 121 35 131)(26 132 36 122)(27 123 37 133)(28 134 38 124)(29 125 39 135)(30 136 40 126)(41 185 51 195)(42 196 52 186)(43 187 53 197)(44 198 54 188)(45 189 55 199)(46 200 56 190)(47 191 57 181)(48 182 58 192)(49 193 59 183)(50 184 60 194)(61 116 71 106)(62 107 72 117)(63 118 73 108)(64 109 74 119)(65 120 75 110)(66 111 76 101)(67 102 77 112)(68 113 78 103)(69 104 79 114)(70 115 80 105)(81 173 91 163)(82 164 92 174)(83 175 93 165)(84 166 94 176)(85 177 95 167)(86 168 96 178)(87 179 97 169)(88 170 98 180)(89 161 99 171)(90 172 100 162)(141 204 151 214)(142 215 152 205)(143 206 153 216)(144 217 154 207)(145 208 155 218)(146 219 156 209)(147 210 157 220)(148 201 158 211)(149 212 159 202)(150 203 160 213)
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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,60)(18,59)(19,58)(20,57)(21,158)(22,157)(23,156)(24,155)(25,154)(26,153)(27,152)(28,151)(29,150)(30,149)(31,148)(32,147)(33,146)(34,145)(35,144)(36,143)(37,142)(38,141)(39,160)(40,159)(61,179)(62,178)(63,177)(64,176)(65,175)(66,174)(67,173)(68,172)(69,171)(70,170)(71,169)(72,168)(73,167)(74,166)(75,165)(76,164)(77,163)(78,162)(79,161)(80,180)(81,102)(82,101)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(121,217)(122,216)(123,215)(124,214)(125,213)(126,212)(127,211)(128,210)(129,209)(130,208)(131,207)(132,206)(133,205)(134,204)(135,203)(136,202)(137,201)(138,220)(139,219)(140,218)(181,226)(182,225)(183,224)(184,223)(185,222)(186,221)(187,240)(188,239)(189,238)(190,237)(191,236)(192,235)(193,234)(194,233)(195,232)(196,231)(197,230)(198,229)(199,228)(200,227), (1,94,151,16,89,146,11,84,141,6,99,156)(2,95,152,17,90,147,12,85,142,7,100,157)(3,96,153,18,91,148,13,86,143,8,81,158)(4,97,154,19,92,149,14,87,144,9,82,159)(5,98,155,20,93,150,15,88,145,10,83,160)(21,54,107,26,59,112,31,44,117,36,49,102)(22,55,108,27,60,113,32,45,118,37,50,103)(23,56,109,28,41,114,33,46,119,38,51,104)(24,57,110,29,42,115,34,47,120,39,52,105)(25,58,111,30,43,116,35,48,101,40,53,106)(61,197,126,66,182,131,71,187,136,76,192,121)(62,198,127,67,183,132,72,188,137,77,193,122)(63,199,128,68,184,133,73,189,138,78,194,123)(64,200,129,69,185,134,74,190,139,79,195,124)(65,181,130,70,186,135,75,191,140,80,196,125)(161,232,214,176,227,209,171,222,204,166,237,219)(162,233,215,177,228,210,172,223,205,167,238,220)(163,234,216,178,229,211,173,224,206,168,239,201)(164,235,217,179,230,212,174,225,207,169,240,202)(165,236,218,180,231,213,175,226,208,170,221,203), (1,227,11,237)(2,238,12,228)(3,229,13,239)(4,240,14,230)(5,231,15,221)(6,222,16,232)(7,233,17,223)(8,224,18,234)(9,235,19,225)(10,226,20,236)(21,137,31,127)(22,128,32,138)(23,139,33,129)(24,130,34,140)(25,121,35,131)(26,132,36,122)(27,123,37,133)(28,134,38,124)(29,125,39,135)(30,136,40,126)(41,185,51,195)(42,196,52,186)(43,187,53,197)(44,198,54,188)(45,189,55,199)(46,200,56,190)(47,191,57,181)(48,182,58,192)(49,193,59,183)(50,184,60,194)(61,116,71,106)(62,107,72,117)(63,118,73,108)(64,109,74,119)(65,120,75,110)(66,111,76,101)(67,102,77,112)(68,113,78,103)(69,104,79,114)(70,115,80,105)(81,173,91,163)(82,164,92,174)(83,175,93,165)(84,166,94,176)(85,177,95,167)(86,168,96,178)(87,179,97,169)(88,170,98,180)(89,161,99,171)(90,172,100,162)(141,204,151,214)(142,215,152,205)(143,206,153,216)(144,217,154,207)(145,208,155,218)(146,219,156,209)(147,210,157,220)(148,201,158,211)(149,212,159,202)(150,203,160,213)>;
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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,60)(18,59)(19,58)(20,57)(21,158)(22,157)(23,156)(24,155)(25,154)(26,153)(27,152)(28,151)(29,150)(30,149)(31,148)(32,147)(33,146)(34,145)(35,144)(36,143)(37,142)(38,141)(39,160)(40,159)(61,179)(62,178)(63,177)(64,176)(65,175)(66,174)(67,173)(68,172)(69,171)(70,170)(71,169)(72,168)(73,167)(74,166)(75,165)(76,164)(77,163)(78,162)(79,161)(80,180)(81,102)(82,101)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(121,217)(122,216)(123,215)(124,214)(125,213)(126,212)(127,211)(128,210)(129,209)(130,208)(131,207)(132,206)(133,205)(134,204)(135,203)(136,202)(137,201)(138,220)(139,219)(140,218)(181,226)(182,225)(183,224)(184,223)(185,222)(186,221)(187,240)(188,239)(189,238)(190,237)(191,236)(192,235)(193,234)(194,233)(195,232)(196,231)(197,230)(198,229)(199,228)(200,227), (1,94,151,16,89,146,11,84,141,6,99,156)(2,95,152,17,90,147,12,85,142,7,100,157)(3,96,153,18,91,148,13,86,143,8,81,158)(4,97,154,19,92,149,14,87,144,9,82,159)(5,98,155,20,93,150,15,88,145,10,83,160)(21,54,107,26,59,112,31,44,117,36,49,102)(22,55,108,27,60,113,32,45,118,37,50,103)(23,56,109,28,41,114,33,46,119,38,51,104)(24,57,110,29,42,115,34,47,120,39,52,105)(25,58,111,30,43,116,35,48,101,40,53,106)(61,197,126,66,182,131,71,187,136,76,192,121)(62,198,127,67,183,132,72,188,137,77,193,122)(63,199,128,68,184,133,73,189,138,78,194,123)(64,200,129,69,185,134,74,190,139,79,195,124)(65,181,130,70,186,135,75,191,140,80,196,125)(161,232,214,176,227,209,171,222,204,166,237,219)(162,233,215,177,228,210,172,223,205,167,238,220)(163,234,216,178,229,211,173,224,206,168,239,201)(164,235,217,179,230,212,174,225,207,169,240,202)(165,236,218,180,231,213,175,226,208,170,221,203), (1,227,11,237)(2,238,12,228)(3,229,13,239)(4,240,14,230)(5,231,15,221)(6,222,16,232)(7,233,17,223)(8,224,18,234)(9,235,19,225)(10,226,20,236)(21,137,31,127)(22,128,32,138)(23,139,33,129)(24,130,34,140)(25,121,35,131)(26,132,36,122)(27,123,37,133)(28,134,38,124)(29,125,39,135)(30,136,40,126)(41,185,51,195)(42,196,52,186)(43,187,53,197)(44,198,54,188)(45,189,55,199)(46,200,56,190)(47,191,57,181)(48,182,58,192)(49,193,59,183)(50,184,60,194)(61,116,71,106)(62,107,72,117)(63,118,73,108)(64,109,74,119)(65,120,75,110)(66,111,76,101)(67,102,77,112)(68,113,78,103)(69,104,79,114)(70,115,80,105)(81,173,91,163)(82,164,92,174)(83,175,93,165)(84,166,94,176)(85,177,95,167)(86,168,96,178)(87,179,97,169)(88,170,98,180)(89,161,99,171)(90,172,100,162)(141,204,151,214)(142,215,152,205)(143,206,153,216)(144,217,154,207)(145,208,155,218)(146,219,156,209)(147,210,157,220)(148,201,158,211)(149,212,159,202)(150,203,160,213) );
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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,42),(16,41),(17,60),(18,59),(19,58),(20,57),(21,158),(22,157),(23,156),(24,155),(25,154),(26,153),(27,152),(28,151),(29,150),(30,149),(31,148),(32,147),(33,146),(34,145),(35,144),(36,143),(37,142),(38,141),(39,160),(40,159),(61,179),(62,178),(63,177),(64,176),(65,175),(66,174),(67,173),(68,172),(69,171),(70,170),(71,169),(72,168),(73,167),(74,166),(75,165),(76,164),(77,163),(78,162),(79,161),(80,180),(81,102),(82,101),(83,120),(84,119),(85,118),(86,117),(87,116),(88,115),(89,114),(90,113),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(121,217),(122,216),(123,215),(124,214),(125,213),(126,212),(127,211),(128,210),(129,209),(130,208),(131,207),(132,206),(133,205),(134,204),(135,203),(136,202),(137,201),(138,220),(139,219),(140,218),(181,226),(182,225),(183,224),(184,223),(185,222),(186,221),(187,240),(188,239),(189,238),(190,237),(191,236),(192,235),(193,234),(194,233),(195,232),(196,231),(197,230),(198,229),(199,228),(200,227)], [(1,94,151,16,89,146,11,84,141,6,99,156),(2,95,152,17,90,147,12,85,142,7,100,157),(3,96,153,18,91,148,13,86,143,8,81,158),(4,97,154,19,92,149,14,87,144,9,82,159),(5,98,155,20,93,150,15,88,145,10,83,160),(21,54,107,26,59,112,31,44,117,36,49,102),(22,55,108,27,60,113,32,45,118,37,50,103),(23,56,109,28,41,114,33,46,119,38,51,104),(24,57,110,29,42,115,34,47,120,39,52,105),(25,58,111,30,43,116,35,48,101,40,53,106),(61,197,126,66,182,131,71,187,136,76,192,121),(62,198,127,67,183,132,72,188,137,77,193,122),(63,199,128,68,184,133,73,189,138,78,194,123),(64,200,129,69,185,134,74,190,139,79,195,124),(65,181,130,70,186,135,75,191,140,80,196,125),(161,232,214,176,227,209,171,222,204,166,237,219),(162,233,215,177,228,210,172,223,205,167,238,220),(163,234,216,178,229,211,173,224,206,168,239,201),(164,235,217,179,230,212,174,225,207,169,240,202),(165,236,218,180,231,213,175,226,208,170,221,203)], [(1,227,11,237),(2,238,12,228),(3,229,13,239),(4,240,14,230),(5,231,15,221),(6,222,16,232),(7,233,17,223),(8,224,18,234),(9,235,19,225),(10,226,20,236),(21,137,31,127),(22,128,32,138),(23,139,33,129),(24,130,34,140),(25,121,35,131),(26,132,36,122),(27,123,37,133),(28,134,38,124),(29,125,39,135),(30,136,40,126),(41,185,51,195),(42,196,52,186),(43,187,53,197),(44,198,54,188),(45,189,55,199),(46,200,56,190),(47,191,57,181),(48,182,58,192),(49,193,59,183),(50,184,60,194),(61,116,71,106),(62,107,72,117),(63,118,73,108),(64,109,74,119),(65,120,75,110),(66,111,76,101),(67,102,77,112),(68,113,78,103),(69,104,79,114),(70,115,80,105),(81,173,91,163),(82,164,92,174),(83,175,93,165),(84,166,94,176),(85,177,95,167),(86,168,96,178),(87,179,97,169),(88,170,98,180),(89,161,99,171),(90,172,100,162),(141,204,151,214),(142,215,152,205),(143,206,153,216),(144,217,154,207),(145,208,155,218),(146,219,156,209),(147,210,157,220),(148,201,158,211),(149,212,159,202),(150,203,160,213)])
Matrix representation ►G ⊆ GL8(𝔽61)
| 17 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 | 0 | 60 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 17 | 24 | 59 | 33 | 0 | 0 | 0 | 0 |
| 40 | 44 | 33 | 51 | 0 | 0 | 0 | 0 |
| 37 | 55 | 41 | 33 | 0 | 0 | 0 | 0 |
| 55 | 44 | 59 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 60 | 1 | 0 | 60 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 59 | 0 | 35 | 50 | 0 | 0 | 0 | 0 |
| 0 | 59 | 26 | 35 | 0 | 0 | 0 | 0 |
| 38 | 59 | 1 | 0 | 0 | 0 | 0 | 0 |
| 38 | 38 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 1 | 60 | 0 |
| 25 | 44 | 41 | 52 | 0 | 0 | 0 | 0 |
| 17 | 9 | 44 | 41 | 0 | 0 | 0 | 0 |
| 52 | 27 | 52 | 17 | 0 | 0 | 0 | 0 |
| 3 | 52 | 44 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 21 | 31 | 21 |
| 0 | 0 | 0 | 0 | 30 | 21 | 21 | 30 |
G:=sub<GL(8,GF(61))| [17,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,17,0,0,0,0,0,0,0,0,1,1,60,0,0,0,0,0,59,60,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0],[17,40,37,55,0,0,0,0,24,44,55,44,0,0,0,0,59,33,41,59,0,0,0,0,33,51,33,20,0,0,0,0,0,0,0,0,1,1,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,1,60,60],[59,0,38,38,0,0,0,0,0,59,59,38,0,0,0,0,35,26,1,0,0,0,0,0,50,35,0,1,0,0,0,0,0,0,0,0,1,1,0,60,0,0,0,0,59,60,1,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0],[25,17,52,3,0,0,0,0,44,9,27,52,0,0,0,0,41,44,52,44,0,0,0,0,52,41,17,36,0,0,0,0,0,0,0,0,52,31,0,30,0,0,0,0,19,9,21,21,0,0,0,0,0,0,31,21,0,0,0,0,0,0,21,30] >;
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | D6 | D10 | D10 | D10 | 2- (1+4) | S3×D5 | Q8○D12 | C2×S3×D5 | C2×S3×D5 | Q8.10D10 | D20.38D6 |
| kernel | D20.38D6 | D5×Dic6 | D20⋊S3 | D15⋊Q8 | C12.28D10 | Dic5.D6 | C3×C4○D20 | C10×Dic6 | D60⋊11C2 | C4○D20 | C2×Dic6 | Dic10 | C4×D5 | D20 | C5⋊D4 | C2×C20 | Dic6 | C2×Dic3 | C2×C12 | C15 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 8 | 4 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
D_{20}._{38}D_6 % in TeX
G:=Group("D20.38D6"); // GroupNames label
G:=SmallGroup(480,1076);
// by ID
G=gap.SmallGroup(480,1076);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^6=d^2=a^10,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=a^10*c^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀