direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×C6.D4, (C22×C6)⋊4C20, C6.62(D4×C10), C24.2(C5×S3), (C22×C30)⋊16C4, (C2×C30).187D4, C30.445(C2×D4), (C23×C6).2C10, (C23×C30).6C2, (C23×C10).3S3, C23⋊3(C5×Dic3), C30⋊12(C22⋊C4), C6.28(C22×C20), C23.37(S3×C10), (C22×C10)⋊9Dic3, C22⋊3(C10×Dic3), (C2×C30).439C23, C30.235(C22×C4), (C22×Dic3)⋊7C10, (C22×C10).127D6, (C10×Dic3)⋊34C22, C10.51(C22×Dic3), (C22×C30).179C22, (C2×C6)⋊8(C2×C20), C6⋊2(C5×C22⋊C4), C3⋊3(C10×C22⋊C4), (C2×C30)⋊44(C2×C4), C15⋊22(C2×C22⋊C4), (C2×C6).44(C5×D4), C2.4(C10×C3⋊D4), C2.9(Dic3×C2×C10), (Dic3×C2×C10)⋊18C2, C22.27(S3×C2×C10), (C2×Dic3)⋊7(C2×C10), (C2×C10)⋊14(C2×Dic3), C10.147(C2×C3⋊D4), C22.25(C5×C3⋊D4), (C2×C10).97(C3⋊D4), (C22×C6).41(C2×C10), (C2×C6).60(C22×C10), (C2×C10).373(C22×S3), SmallGroup(480,831)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C6.D4
G = < a,b,c,d | a10=b6=c4=1, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >
Subgroups: 484 in 264 conjugacy classes, 130 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C23, C23, C23, C10, C10, C10, Dic3, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C22×C6, C22×C6, C22×C6, C30, C30, C30, C2×C22⋊C4, C2×C20, C22×C10, C22×C10, C22×C10, C6.D4, C22×Dic3, C23×C6, C5×Dic3, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C22×C20, C23×C10, C2×C6.D4, C10×Dic3, C10×Dic3, C22×C30, C22×C30, C22×C30, C10×C22⋊C4, C5×C6.D4, Dic3×C2×C10, C23×C30, C10×C6.D4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C23, C10, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×Dic3, C3⋊D4, C22×S3, C5×S3, C2×C22⋊C4, C2×C20, C5×D4, C22×C10, C6.D4, C22×Dic3, C2×C3⋊D4, C5×Dic3, S3×C10, C5×C22⋊C4, C22×C20, D4×C10, C2×C6.D4, C10×Dic3, C5×C3⋊D4, S3×C2×C10, C10×C22⋊C4, C5×C6.D4, Dic3×C2×C10, C10×C3⋊D4, C10×C6.D4
►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)
(1 157 190 74 195 221)(2 158 181 75 196 222)(3 159 182 76 197 223)(4 160 183 77 198 224)(5 151 184 78 199 225)(6 152 185 79 200 226)(7 153 186 80 191 227)(8 154 187 71 192 228)(9 155 188 72 193 229)(10 156 189 73 194 230)(11 64 45 111 134 231)(12 65 46 112 135 232)(13 66 47 113 136 233)(14 67 48 114 137 234)(15 68 49 115 138 235)(16 69 50 116 139 236)(17 70 41 117 140 237)(18 61 42 118 131 238)(19 62 43 119 132 239)(20 63 44 120 133 240)(21 109 38 95 214 88)(22 110 39 96 215 89)(23 101 40 97 216 90)(24 102 31 98 217 81)(25 103 32 99 218 82)(26 104 33 100 219 83)(27 105 34 91 220 84)(28 106 35 92 211 85)(29 107 36 93 212 86)(30 108 37 94 213 87)(51 161 144 209 180 124)(52 162 145 210 171 125)(53 163 146 201 172 126)(54 164 147 202 173 127)(55 165 148 203 174 128)(56 166 149 204 175 129)(57 167 150 205 176 130)(58 168 141 206 177 121)(59 169 142 207 178 122)(60 170 143 208 179 123)
(1 14 172 101)(2 15 173 102)(3 16 174 103)(4 17 175 104)(5 18 176 105)(6 19 177 106)(7 20 178 107)(8 11 179 108)(9 12 180 109)(10 13 171 110)(21 155 232 124)(22 156 233 125)(23 157 234 126)(24 158 235 127)(25 159 236 128)(26 160 237 129)(27 151 238 130)(28 152 239 121)(29 153 240 122)(30 154 231 123)(31 222 68 202)(32 223 69 203)(33 224 70 204)(34 225 61 205)(35 226 62 206)(36 227 63 207)(37 228 64 208)(38 229 65 209)(39 230 66 210)(40 221 67 201)(41 149 100 198)(42 150 91 199)(43 141 92 200)(44 142 93 191)(45 143 94 192)(46 144 95 193)(47 145 96 194)(48 146 97 195)(49 147 98 196)(50 148 99 197)(51 88 188 135)(52 89 189 136)(53 90 190 137)(54 81 181 138)(55 82 182 139)(56 83 183 140)(57 84 184 131)(58 85 185 132)(59 86 186 133)(60 87 187 134)(71 111 170 213)(72 112 161 214)(73 113 162 215)(74 114 163 216)(75 115 164 217)(76 116 165 218)(77 117 166 219)(78 118 167 220)(79 119 168 211)(80 120 169 212)
(1 216 74 101)(2 217 75 102)(3 218 76 103)(4 219 77 104)(5 220 78 105)(6 211 79 106)(7 212 80 107)(8 213 71 108)(9 214 72 109)(10 215 73 110)(11 179 111 170)(12 180 112 161)(13 171 113 162)(14 172 114 163)(15 173 115 164)(16 174 116 165)(17 175 117 166)(18 176 118 167)(19 177 119 168)(20 178 120 169)(21 155 95 193)(22 156 96 194)(23 157 97 195)(24 158 98 196)(25 159 99 197)(26 160 100 198)(27 151 91 199)(28 152 92 200)(29 153 93 191)(30 154 94 192)(31 222 81 181)(32 223 82 182)(33 224 83 183)(34 225 84 184)(35 226 85 185)(36 227 86 186)(37 228 87 187)(38 229 88 188)(39 230 89 189)(40 221 90 190)(41 149 237 129)(42 150 238 130)(43 141 239 121)(44 142 240 122)(45 143 231 123)(46 144 232 124)(47 145 233 125)(48 146 234 126)(49 147 235 127)(50 148 236 128)(51 65 209 135)(52 66 210 136)(53 67 201 137)(54 68 202 138)(55 69 203 139)(56 70 204 140)(57 61 205 131)(58 62 206 132)(59 63 207 133)(60 64 208 134)
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), (1,157,190,74,195,221)(2,158,181,75,196,222)(3,159,182,76,197,223)(4,160,183,77,198,224)(5,151,184,78,199,225)(6,152,185,79,200,226)(7,153,186,80,191,227)(8,154,187,71,192,228)(9,155,188,72,193,229)(10,156,189,73,194,230)(11,64,45,111,134,231)(12,65,46,112,135,232)(13,66,47,113,136,233)(14,67,48,114,137,234)(15,68,49,115,138,235)(16,69,50,116,139,236)(17,70,41,117,140,237)(18,61,42,118,131,238)(19,62,43,119,132,239)(20,63,44,120,133,240)(21,109,38,95,214,88)(22,110,39,96,215,89)(23,101,40,97,216,90)(24,102,31,98,217,81)(25,103,32,99,218,82)(26,104,33,100,219,83)(27,105,34,91,220,84)(28,106,35,92,211,85)(29,107,36,93,212,86)(30,108,37,94,213,87)(51,161,144,209,180,124)(52,162,145,210,171,125)(53,163,146,201,172,126)(54,164,147,202,173,127)(55,165,148,203,174,128)(56,166,149,204,175,129)(57,167,150,205,176,130)(58,168,141,206,177,121)(59,169,142,207,178,122)(60,170,143,208,179,123), (1,14,172,101)(2,15,173,102)(3,16,174,103)(4,17,175,104)(5,18,176,105)(6,19,177,106)(7,20,178,107)(8,11,179,108)(9,12,180,109)(10,13,171,110)(21,155,232,124)(22,156,233,125)(23,157,234,126)(24,158,235,127)(25,159,236,128)(26,160,237,129)(27,151,238,130)(28,152,239,121)(29,153,240,122)(30,154,231,123)(31,222,68,202)(32,223,69,203)(33,224,70,204)(34,225,61,205)(35,226,62,206)(36,227,63,207)(37,228,64,208)(38,229,65,209)(39,230,66,210)(40,221,67,201)(41,149,100,198)(42,150,91,199)(43,141,92,200)(44,142,93,191)(45,143,94,192)(46,144,95,193)(47,145,96,194)(48,146,97,195)(49,147,98,196)(50,148,99,197)(51,88,188,135)(52,89,189,136)(53,90,190,137)(54,81,181,138)(55,82,182,139)(56,83,183,140)(57,84,184,131)(58,85,185,132)(59,86,186,133)(60,87,187,134)(71,111,170,213)(72,112,161,214)(73,113,162,215)(74,114,163,216)(75,115,164,217)(76,116,165,218)(77,117,166,219)(78,118,167,220)(79,119,168,211)(80,120,169,212), (1,216,74,101)(2,217,75,102)(3,218,76,103)(4,219,77,104)(5,220,78,105)(6,211,79,106)(7,212,80,107)(8,213,71,108)(9,214,72,109)(10,215,73,110)(11,179,111,170)(12,180,112,161)(13,171,113,162)(14,172,114,163)(15,173,115,164)(16,174,116,165)(17,175,117,166)(18,176,118,167)(19,177,119,168)(20,178,120,169)(21,155,95,193)(22,156,96,194)(23,157,97,195)(24,158,98,196)(25,159,99,197)(26,160,100,198)(27,151,91,199)(28,152,92,200)(29,153,93,191)(30,154,94,192)(31,222,81,181)(32,223,82,182)(33,224,83,183)(34,225,84,184)(35,226,85,185)(36,227,86,186)(37,228,87,187)(38,229,88,188)(39,230,89,189)(40,221,90,190)(41,149,237,129)(42,150,238,130)(43,141,239,121)(44,142,240,122)(45,143,231,123)(46,144,232,124)(47,145,233,125)(48,146,234,126)(49,147,235,127)(50,148,236,128)(51,65,209,135)(52,66,210,136)(53,67,201,137)(54,68,202,138)(55,69,203,139)(56,70,204,140)(57,61,205,131)(58,62,206,132)(59,63,207,133)(60,64,208,134)>;
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), (1,157,190,74,195,221)(2,158,181,75,196,222)(3,159,182,76,197,223)(4,160,183,77,198,224)(5,151,184,78,199,225)(6,152,185,79,200,226)(7,153,186,80,191,227)(8,154,187,71,192,228)(9,155,188,72,193,229)(10,156,189,73,194,230)(11,64,45,111,134,231)(12,65,46,112,135,232)(13,66,47,113,136,233)(14,67,48,114,137,234)(15,68,49,115,138,235)(16,69,50,116,139,236)(17,70,41,117,140,237)(18,61,42,118,131,238)(19,62,43,119,132,239)(20,63,44,120,133,240)(21,109,38,95,214,88)(22,110,39,96,215,89)(23,101,40,97,216,90)(24,102,31,98,217,81)(25,103,32,99,218,82)(26,104,33,100,219,83)(27,105,34,91,220,84)(28,106,35,92,211,85)(29,107,36,93,212,86)(30,108,37,94,213,87)(51,161,144,209,180,124)(52,162,145,210,171,125)(53,163,146,201,172,126)(54,164,147,202,173,127)(55,165,148,203,174,128)(56,166,149,204,175,129)(57,167,150,205,176,130)(58,168,141,206,177,121)(59,169,142,207,178,122)(60,170,143,208,179,123), (1,14,172,101)(2,15,173,102)(3,16,174,103)(4,17,175,104)(5,18,176,105)(6,19,177,106)(7,20,178,107)(8,11,179,108)(9,12,180,109)(10,13,171,110)(21,155,232,124)(22,156,233,125)(23,157,234,126)(24,158,235,127)(25,159,236,128)(26,160,237,129)(27,151,238,130)(28,152,239,121)(29,153,240,122)(30,154,231,123)(31,222,68,202)(32,223,69,203)(33,224,70,204)(34,225,61,205)(35,226,62,206)(36,227,63,207)(37,228,64,208)(38,229,65,209)(39,230,66,210)(40,221,67,201)(41,149,100,198)(42,150,91,199)(43,141,92,200)(44,142,93,191)(45,143,94,192)(46,144,95,193)(47,145,96,194)(48,146,97,195)(49,147,98,196)(50,148,99,197)(51,88,188,135)(52,89,189,136)(53,90,190,137)(54,81,181,138)(55,82,182,139)(56,83,183,140)(57,84,184,131)(58,85,185,132)(59,86,186,133)(60,87,187,134)(71,111,170,213)(72,112,161,214)(73,113,162,215)(74,114,163,216)(75,115,164,217)(76,116,165,218)(77,117,166,219)(78,118,167,220)(79,119,168,211)(80,120,169,212), (1,216,74,101)(2,217,75,102)(3,218,76,103)(4,219,77,104)(5,220,78,105)(6,211,79,106)(7,212,80,107)(8,213,71,108)(9,214,72,109)(10,215,73,110)(11,179,111,170)(12,180,112,161)(13,171,113,162)(14,172,114,163)(15,173,115,164)(16,174,116,165)(17,175,117,166)(18,176,118,167)(19,177,119,168)(20,178,120,169)(21,155,95,193)(22,156,96,194)(23,157,97,195)(24,158,98,196)(25,159,99,197)(26,160,100,198)(27,151,91,199)(28,152,92,200)(29,153,93,191)(30,154,94,192)(31,222,81,181)(32,223,82,182)(33,224,83,183)(34,225,84,184)(35,226,85,185)(36,227,86,186)(37,228,87,187)(38,229,88,188)(39,230,89,189)(40,221,90,190)(41,149,237,129)(42,150,238,130)(43,141,239,121)(44,142,240,122)(45,143,231,123)(46,144,232,124)(47,145,233,125)(48,146,234,126)(49,147,235,127)(50,148,236,128)(51,65,209,135)(52,66,210,136)(53,67,201,137)(54,68,202,138)(55,69,203,139)(56,70,204,140)(57,61,205,131)(58,62,206,132)(59,63,207,133)(60,64,208,134) );
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)], [(1,157,190,74,195,221),(2,158,181,75,196,222),(3,159,182,76,197,223),(4,160,183,77,198,224),(5,151,184,78,199,225),(6,152,185,79,200,226),(7,153,186,80,191,227),(8,154,187,71,192,228),(9,155,188,72,193,229),(10,156,189,73,194,230),(11,64,45,111,134,231),(12,65,46,112,135,232),(13,66,47,113,136,233),(14,67,48,114,137,234),(15,68,49,115,138,235),(16,69,50,116,139,236),(17,70,41,117,140,237),(18,61,42,118,131,238),(19,62,43,119,132,239),(20,63,44,120,133,240),(21,109,38,95,214,88),(22,110,39,96,215,89),(23,101,40,97,216,90),(24,102,31,98,217,81),(25,103,32,99,218,82),(26,104,33,100,219,83),(27,105,34,91,220,84),(28,106,35,92,211,85),(29,107,36,93,212,86),(30,108,37,94,213,87),(51,161,144,209,180,124),(52,162,145,210,171,125),(53,163,146,201,172,126),(54,164,147,202,173,127),(55,165,148,203,174,128),(56,166,149,204,175,129),(57,167,150,205,176,130),(58,168,141,206,177,121),(59,169,142,207,178,122),(60,170,143,208,179,123)], [(1,14,172,101),(2,15,173,102),(3,16,174,103),(4,17,175,104),(5,18,176,105),(6,19,177,106),(7,20,178,107),(8,11,179,108),(9,12,180,109),(10,13,171,110),(21,155,232,124),(22,156,233,125),(23,157,234,126),(24,158,235,127),(25,159,236,128),(26,160,237,129),(27,151,238,130),(28,152,239,121),(29,153,240,122),(30,154,231,123),(31,222,68,202),(32,223,69,203),(33,224,70,204),(34,225,61,205),(35,226,62,206),(36,227,63,207),(37,228,64,208),(38,229,65,209),(39,230,66,210),(40,221,67,201),(41,149,100,198),(42,150,91,199),(43,141,92,200),(44,142,93,191),(45,143,94,192),(46,144,95,193),(47,145,96,194),(48,146,97,195),(49,147,98,196),(50,148,99,197),(51,88,188,135),(52,89,189,136),(53,90,190,137),(54,81,181,138),(55,82,182,139),(56,83,183,140),(57,84,184,131),(58,85,185,132),(59,86,186,133),(60,87,187,134),(71,111,170,213),(72,112,161,214),(73,113,162,215),(74,114,163,216),(75,115,164,217),(76,116,165,218),(77,117,166,219),(78,118,167,220),(79,119,168,211),(80,120,169,212)], [(1,216,74,101),(2,217,75,102),(3,218,76,103),(4,219,77,104),(5,220,78,105),(6,211,79,106),(7,212,80,107),(8,213,71,108),(9,214,72,109),(10,215,73,110),(11,179,111,170),(12,180,112,161),(13,171,113,162),(14,172,114,163),(15,173,115,164),(16,174,116,165),(17,175,117,166),(18,176,118,167),(19,177,119,168),(20,178,120,169),(21,155,95,193),(22,156,96,194),(23,157,97,195),(24,158,98,196),(25,159,99,197),(26,160,100,198),(27,151,91,199),(28,152,92,200),(29,153,93,191),(30,154,94,192),(31,222,81,181),(32,223,82,182),(33,224,83,183),(34,225,84,184),(35,226,85,185),(36,227,86,186),(37,228,87,187),(38,229,88,188),(39,230,89,189),(40,221,90,190),(41,149,237,129),(42,150,238,130),(43,141,239,121),(44,142,240,122),(45,143,231,123),(46,144,232,124),(47,145,233,125),(48,146,234,126),(49,147,235,127),(50,148,236,128),(51,65,209,135),(52,66,210,136),(53,67,201,137),(54,68,202,138),(55,69,203,139),(56,70,204,140),(57,61,205,131),(58,62,206,132),(59,63,207,133),(60,64,208,134)]])
180 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4H | 5A | 5B | 5C | 5D | 6A | ··· | 6O | 10A | ··· | 10AB | 10AC | ··· | 10AR | 15A | 15B | 15C | 15D | 20A | ··· | 20AF | 30A | ··· | 30BH |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D4 | Dic3 | D6 | C3⋊D4 | C5×S3 | C5×D4 | C5×Dic3 | S3×C10 | C5×C3⋊D4 |
| kernel | C10×C6.D4 | C5×C6.D4 | Dic3×C2×C10 | C23×C30 | C22×C30 | C2×C6.D4 | C6.D4 | C22×Dic3 | C23×C6 | C22×C6 | C23×C10 | C2×C30 | C22×C10 | C22×C10 | C2×C10 | C24 | C2×C6 | C23 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 4 | 16 | 8 | 4 | 32 | 1 | 4 | 4 | 3 | 8 | 4 | 16 | 16 | 12 | 32 |
Matrix representation of C10×C6.D4 ►in GL6(𝔽61)
| 52 | 0 | 0 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 0 | 0 | 52 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 1 | 59 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 1 | 59 | 0 | 0 | 0 | 0 |
| 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(61))| [52,0,0,0,0,0,0,52,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,52,0,0,0,0,0,0,52],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,47,0,0,0,0,0,0,13],[1,0,0,0,0,0,59,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,1,0],[1,1,0,0,0,0,59,60,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C10×C6.D4 in GAP, Magma, Sage, TeX
C_{10}\times C_6.D_4 % in TeX
G:=Group("C10xC6.D4"); // GroupNames label
G:=SmallGroup(480,831);
// by ID
G=gap.SmallGroup(480,831);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,1766,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^6=c^4=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
// generators/relations