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## G = C10×D6⋊C4order 480 = 25·3·5

### Direct product of C10 and D6⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C10×D6⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — S3×C22×C10 — C10×D6⋊C4
 Lower central C3 — C6 — C10×D6⋊C4
 Upper central C1 — C22×C10 — C22×C20

Generators and relations for C10×D6⋊C4
G = < a,b,c,d | a10=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 644 in 264 conjugacy classes, 114 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C22, C22 [×6], C22 [×16], C5, S3 [×4], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], C10 [×3], C10 [×4], C10 [×4], Dic3 [×2], C12 [×2], D6 [×4], D6 [×12], C2×C6, C2×C6 [×6], C15, C22⋊C4 [×4], C22×C4, C22×C4, C24, C20 [×4], C2×C10, C2×C10 [×6], C2×C10 [×16], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×2], C22×S3 [×6], C22×S3 [×4], C22×C6, C5×S3 [×4], C30 [×3], C30 [×4], C2×C22⋊C4, C2×C20 [×2], C2×C20 [×6], C22×C10, C22×C10 [×10], D6⋊C4 [×4], C22×Dic3, C22×C12, S3×C23, C5×Dic3 [×2], C60 [×2], S3×C10 [×4], S3×C10 [×12], C2×C30, C2×C30 [×6], C5×C22⋊C4 [×4], C22×C20, C22×C20, C23×C10, C2×D6⋊C4, C10×Dic3 [×2], C10×Dic3 [×2], C2×C60 [×2], C2×C60 [×2], S3×C2×C10 [×6], S3×C2×C10 [×4], C22×C30, C10×C22⋊C4, C5×D6⋊C4 [×4], Dic3×C2×C10, C22×C60, S3×C22×C10, C10×D6⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], D4 [×4], C23, C10 [×7], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C5×S3, C2×C22⋊C4, C2×C20 [×6], C5×D4 [×4], C22×C10, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, S3×C10 [×3], C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], C2×D6⋊C4, S3×C20 [×2], C5×D12 [×2], C5×C3⋊D4 [×2], S3×C2×C10, C10×C22⋊C4, C5×D6⋊C4 [×4], S3×C2×C20, C10×D12, C10×C3⋊D4, C10×D6⋊C4

Smallest permutation representation of C10×D6⋊C4
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 55 105 188 16 72)(2 56 106 189 17 73)(3 57 107 190 18 74)(4 58 108 181 19 75)(5 59 109 182 20 76)(6 60 110 183 11 77)(7 51 101 184 12 78)(8 52 102 185 13 79)(9 53 103 186 14 80)(10 54 104 187 15 71)(21 31 206 197 98 41)(22 32 207 198 99 42)(23 33 208 199 100 43)(24 34 209 200 91 44)(25 35 210 191 92 45)(26 36 201 192 93 46)(27 37 202 193 94 47)(28 38 203 194 95 48)(29 39 204 195 96 49)(30 40 205 196 97 50)(61 139 234 150 162 90)(62 140 235 141 163 81)(63 131 236 142 164 82)(64 132 237 143 165 83)(65 133 238 144 166 84)(66 134 239 145 167 85)(67 135 240 146 168 86)(68 136 231 147 169 87)(69 137 232 148 170 88)(70 138 233 149 161 89)(111 222 215 153 171 121)(112 223 216 154 172 122)(113 224 217 155 173 123)(114 225 218 156 174 124)(115 226 219 157 175 125)(116 227 220 158 176 126)(117 228 211 159 177 127)(118 229 212 160 178 128)(119 230 213 151 179 129)(120 221 214 152 180 130)
(1 110)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 16)(12 17)(13 18)(14 19)(15 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 201)(42 202)(43 203)(44 204)(45 205)(46 206)(47 207)(48 208)(49 209)(50 210)(51 56)(52 57)(53 58)(54 59)(55 60)(61 167)(62 168)(63 169)(64 170)(65 161)(66 162)(67 163)(68 164)(69 165)(70 166)(71 182)(72 183)(73 184)(74 185)(75 186)(76 187)(77 188)(78 189)(79 190)(80 181)(81 86)(82 87)(83 88)(84 89)(85 90)(91 195)(92 196)(93 197)(94 198)(95 199)(96 200)(97 191)(98 192)(99 193)(100 194)(111 158)(112 159)(113 160)(114 151)(115 152)(116 153)(117 154)(118 155)(119 156)(120 157)(121 176)(122 177)(123 178)(124 179)(125 180)(126 171)(127 172)(128 173)(129 174)(130 175)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)(211 223)(212 224)(213 225)(214 226)(215 227)(216 228)(217 229)(218 230)(219 221)(220 222)(231 236)(232 237)(233 238)(234 239)(235 240)
(1 38 167 125)(2 39 168 126)(3 40 169 127)(4 31 170 128)(5 32 161 129)(6 33 162 130)(7 34 163 121)(8 35 164 122)(9 36 165 123)(10 37 166 124)(11 43 234 152)(12 44 235 153)(13 45 236 154)(14 46 237 155)(15 47 238 156)(16 48 239 157)(17 49 240 158)(18 50 231 159)(19 41 232 160)(20 42 233 151)(21 148 178 75)(22 149 179 76)(23 150 180 77)(24 141 171 78)(25 142 172 79)(26 143 173 80)(27 144 174 71)(28 145 175 72)(29 146 176 73)(30 147 177 74)(51 209 81 111)(52 210 82 112)(53 201 83 113)(54 202 84 114)(55 203 85 115)(56 204 86 116)(57 205 87 117)(58 206 88 118)(59 207 89 119)(60 208 90 120)(61 221 110 199)(62 222 101 200)(63 223 102 191)(64 224 103 192)(65 225 104 193)(66 226 105 194)(67 227 106 195)(68 228 107 196)(69 229 108 197)(70 230 109 198)(91 140 215 184)(92 131 216 185)(93 132 217 186)(94 133 218 187)(95 134 219 188)(96 135 220 189)(97 136 211 190)(98 137 212 181)(99 138 213 182)(100 139 214 183)

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), (1,55,105,188,16,72)(2,56,106,189,17,73)(3,57,107,190,18,74)(4,58,108,181,19,75)(5,59,109,182,20,76)(6,60,110,183,11,77)(7,51,101,184,12,78)(8,52,102,185,13,79)(9,53,103,186,14,80)(10,54,104,187,15,71)(21,31,206,197,98,41)(22,32,207,198,99,42)(23,33,208,199,100,43)(24,34,209,200,91,44)(25,35,210,191,92,45)(26,36,201,192,93,46)(27,37,202,193,94,47)(28,38,203,194,95,48)(29,39,204,195,96,49)(30,40,205,196,97,50)(61,139,234,150,162,90)(62,140,235,141,163,81)(63,131,236,142,164,82)(64,132,237,143,165,83)(65,133,238,144,166,84)(66,134,239,145,167,85)(67,135,240,146,168,86)(68,136,231,147,169,87)(69,137,232,148,170,88)(70,138,233,149,161,89)(111,222,215,153,171,121)(112,223,216,154,172,122)(113,224,217,155,173,123)(114,225,218,156,174,124)(115,226,219,157,175,125)(116,227,220,158,176,126)(117,228,211,159,177,127)(118,229,212,160,178,128)(119,230,213,151,179,129)(120,221,214,152,180,130), (1,110)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,16)(12,17)(13,18)(14,19)(15,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35)(41,201)(42,202)(43,203)(44,204)(45,205)(46,206)(47,207)(48,208)(49,209)(50,210)(51,56)(52,57)(53,58)(54,59)(55,60)(61,167)(62,168)(63,169)(64,170)(65,161)(66,162)(67,163)(68,164)(69,165)(70,166)(71,182)(72,183)(73,184)(74,185)(75,186)(76,187)(77,188)(78,189)(79,190)(80,181)(81,86)(82,87)(83,88)(84,89)(85,90)(91,195)(92,196)(93,197)(94,198)(95,199)(96,200)(97,191)(98,192)(99,193)(100,194)(111,158)(112,159)(113,160)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,157)(121,176)(122,177)(123,178)(124,179)(125,180)(126,171)(127,172)(128,173)(129,174)(130,175)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(211,223)(212,224)(213,225)(214,226)(215,227)(216,228)(217,229)(218,230)(219,221)(220,222)(231,236)(232,237)(233,238)(234,239)(235,240), (1,38,167,125)(2,39,168,126)(3,40,169,127)(4,31,170,128)(5,32,161,129)(6,33,162,130)(7,34,163,121)(8,35,164,122)(9,36,165,123)(10,37,166,124)(11,43,234,152)(12,44,235,153)(13,45,236,154)(14,46,237,155)(15,47,238,156)(16,48,239,157)(17,49,240,158)(18,50,231,159)(19,41,232,160)(20,42,233,151)(21,148,178,75)(22,149,179,76)(23,150,180,77)(24,141,171,78)(25,142,172,79)(26,143,173,80)(27,144,174,71)(28,145,175,72)(29,146,176,73)(30,147,177,74)(51,209,81,111)(52,210,82,112)(53,201,83,113)(54,202,84,114)(55,203,85,115)(56,204,86,116)(57,205,87,117)(58,206,88,118)(59,207,89,119)(60,208,90,120)(61,221,110,199)(62,222,101,200)(63,223,102,191)(64,224,103,192)(65,225,104,193)(66,226,105,194)(67,227,106,195)(68,228,107,196)(69,229,108,197)(70,230,109,198)(91,140,215,184)(92,131,216,185)(93,132,217,186)(94,133,218,187)(95,134,219,188)(96,135,220,189)(97,136,211,190)(98,137,212,181)(99,138,213,182)(100,139,214,183)>;

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), (1,55,105,188,16,72)(2,56,106,189,17,73)(3,57,107,190,18,74)(4,58,108,181,19,75)(5,59,109,182,20,76)(6,60,110,183,11,77)(7,51,101,184,12,78)(8,52,102,185,13,79)(9,53,103,186,14,80)(10,54,104,187,15,71)(21,31,206,197,98,41)(22,32,207,198,99,42)(23,33,208,199,100,43)(24,34,209,200,91,44)(25,35,210,191,92,45)(26,36,201,192,93,46)(27,37,202,193,94,47)(28,38,203,194,95,48)(29,39,204,195,96,49)(30,40,205,196,97,50)(61,139,234,150,162,90)(62,140,235,141,163,81)(63,131,236,142,164,82)(64,132,237,143,165,83)(65,133,238,144,166,84)(66,134,239,145,167,85)(67,135,240,146,168,86)(68,136,231,147,169,87)(69,137,232,148,170,88)(70,138,233,149,161,89)(111,222,215,153,171,121)(112,223,216,154,172,122)(113,224,217,155,173,123)(114,225,218,156,174,124)(115,226,219,157,175,125)(116,227,220,158,176,126)(117,228,211,159,177,127)(118,229,212,160,178,128)(119,230,213,151,179,129)(120,221,214,152,180,130), (1,110)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,16)(12,17)(13,18)(14,19)(15,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35)(41,201)(42,202)(43,203)(44,204)(45,205)(46,206)(47,207)(48,208)(49,209)(50,210)(51,56)(52,57)(53,58)(54,59)(55,60)(61,167)(62,168)(63,169)(64,170)(65,161)(66,162)(67,163)(68,164)(69,165)(70,166)(71,182)(72,183)(73,184)(74,185)(75,186)(76,187)(77,188)(78,189)(79,190)(80,181)(81,86)(82,87)(83,88)(84,89)(85,90)(91,195)(92,196)(93,197)(94,198)(95,199)(96,200)(97,191)(98,192)(99,193)(100,194)(111,158)(112,159)(113,160)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,157)(121,176)(122,177)(123,178)(124,179)(125,180)(126,171)(127,172)(128,173)(129,174)(130,175)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(211,223)(212,224)(213,225)(214,226)(215,227)(216,228)(217,229)(218,230)(219,221)(220,222)(231,236)(232,237)(233,238)(234,239)(235,240), (1,38,167,125)(2,39,168,126)(3,40,169,127)(4,31,170,128)(5,32,161,129)(6,33,162,130)(7,34,163,121)(8,35,164,122)(9,36,165,123)(10,37,166,124)(11,43,234,152)(12,44,235,153)(13,45,236,154)(14,46,237,155)(15,47,238,156)(16,48,239,157)(17,49,240,158)(18,50,231,159)(19,41,232,160)(20,42,233,151)(21,148,178,75)(22,149,179,76)(23,150,180,77)(24,141,171,78)(25,142,172,79)(26,143,173,80)(27,144,174,71)(28,145,175,72)(29,146,176,73)(30,147,177,74)(51,209,81,111)(52,210,82,112)(53,201,83,113)(54,202,84,114)(55,203,85,115)(56,204,86,116)(57,205,87,117)(58,206,88,118)(59,207,89,119)(60,208,90,120)(61,221,110,199)(62,222,101,200)(63,223,102,191)(64,224,103,192)(65,225,104,193)(66,226,105,194)(67,227,106,195)(68,228,107,196)(69,229,108,197)(70,230,109,198)(91,140,215,184)(92,131,216,185)(93,132,217,186)(94,133,218,187)(95,134,219,188)(96,135,220,189)(97,136,211,190)(98,137,212,181)(99,138,213,182)(100,139,214,183) );

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)], [(1,55,105,188,16,72),(2,56,106,189,17,73),(3,57,107,190,18,74),(4,58,108,181,19,75),(5,59,109,182,20,76),(6,60,110,183,11,77),(7,51,101,184,12,78),(8,52,102,185,13,79),(9,53,103,186,14,80),(10,54,104,187,15,71),(21,31,206,197,98,41),(22,32,207,198,99,42),(23,33,208,199,100,43),(24,34,209,200,91,44),(25,35,210,191,92,45),(26,36,201,192,93,46),(27,37,202,193,94,47),(28,38,203,194,95,48),(29,39,204,195,96,49),(30,40,205,196,97,50),(61,139,234,150,162,90),(62,140,235,141,163,81),(63,131,236,142,164,82),(64,132,237,143,165,83),(65,133,238,144,166,84),(66,134,239,145,167,85),(67,135,240,146,168,86),(68,136,231,147,169,87),(69,137,232,148,170,88),(70,138,233,149,161,89),(111,222,215,153,171,121),(112,223,216,154,172,122),(113,224,217,155,173,123),(114,225,218,156,174,124),(115,226,219,157,175,125),(116,227,220,158,176,126),(117,228,211,159,177,127),(118,229,212,160,178,128),(119,230,213,151,179,129),(120,221,214,152,180,130)], [(1,110),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,109),(11,16),(12,17),(13,18),(14,19),(15,20),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35),(41,201),(42,202),(43,203),(44,204),(45,205),(46,206),(47,207),(48,208),(49,209),(50,210),(51,56),(52,57),(53,58),(54,59),(55,60),(61,167),(62,168),(63,169),(64,170),(65,161),(66,162),(67,163),(68,164),(69,165),(70,166),(71,182),(72,183),(73,184),(74,185),(75,186),(76,187),(77,188),(78,189),(79,190),(80,181),(81,86),(82,87),(83,88),(84,89),(85,90),(91,195),(92,196),(93,197),(94,198),(95,199),(96,200),(97,191),(98,192),(99,193),(100,194),(111,158),(112,159),(113,160),(114,151),(115,152),(116,153),(117,154),(118,155),(119,156),(120,157),(121,176),(122,177),(123,178),(124,179),(125,180),(126,171),(127,172),(128,173),(129,174),(130,175),(131,147),(132,148),(133,149),(134,150),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146),(211,223),(212,224),(213,225),(214,226),(215,227),(216,228),(217,229),(218,230),(219,221),(220,222),(231,236),(232,237),(233,238),(234,239),(235,240)], [(1,38,167,125),(2,39,168,126),(3,40,169,127),(4,31,170,128),(5,32,161,129),(6,33,162,130),(7,34,163,121),(8,35,164,122),(9,36,165,123),(10,37,166,124),(11,43,234,152),(12,44,235,153),(13,45,236,154),(14,46,237,155),(15,47,238,156),(16,48,239,157),(17,49,240,158),(18,50,231,159),(19,41,232,160),(20,42,233,151),(21,148,178,75),(22,149,179,76),(23,150,180,77),(24,141,171,78),(25,142,172,79),(26,143,173,80),(27,144,174,71),(28,145,175,72),(29,146,176,73),(30,147,177,74),(51,209,81,111),(52,210,82,112),(53,201,83,113),(54,202,84,114),(55,203,85,115),(56,204,86,116),(57,205,87,117),(58,206,88,118),(59,207,89,119),(60,208,90,120),(61,221,110,199),(62,222,101,200),(63,223,102,191),(64,224,103,192),(65,225,104,193),(66,226,105,194),(67,227,106,195),(68,228,107,196),(69,229,108,197),(70,230,109,198),(91,140,215,184),(92,131,216,185),(93,132,217,186),(94,133,218,187),(95,134,219,188),(96,135,220,189),(97,136,211,190),(98,137,212,181),(99,138,213,182),(100,139,214,183)])

180 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 6A ··· 6G 10A ··· 10AB 10AC ··· 10AR 12A ··· 12H 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20AF 30A ··· 30AB 60A ··· 60AF order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 5 5 6 ··· 6 10 ··· 10 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 6 6 6 6 2 2 2 2 2 6 6 6 6 1 1 1 1 2 ··· 2 1 ··· 1 6 ··· 6 2 ··· 2 2 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C20 S3 D4 D6 D6 C4×S3 D12 C3⋊D4 C5×S3 C5×D4 S3×C10 S3×C10 S3×C20 C5×D12 C5×C3⋊D4 kernel C10×D6⋊C4 C5×D6⋊C4 Dic3×C2×C10 C22×C60 S3×C22×C10 S3×C2×C10 C2×D6⋊C4 D6⋊C4 C22×Dic3 C22×C12 S3×C23 C22×S3 C22×C20 C2×C30 C2×C20 C22×C10 C2×C10 C2×C10 C2×C10 C22×C4 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 4 1 1 1 8 4 16 4 4 4 32 1 4 2 1 4 4 4 4 16 8 4 16 16 16

Matrix representation of C10×D6⋊C4 in GL4(𝔽61) generated by

 60 0 0 0 0 1 0 0 0 0 27 0 0 0 0 27
,
 1 0 0 0 0 1 0 0 0 0 0 60 0 0 1 1
,
 1 0 0 0 0 60 0 0 0 0 0 60 0 0 60 0
,
 1 0 0 0 0 50 0 0 0 0 9 18 0 0 43 52
G:=sub<GL(4,GF(61))| [60,0,0,0,0,1,0,0,0,0,27,0,0,0,0,27],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,1],[1,0,0,0,0,60,0,0,0,0,0,60,0,0,60,0],[1,0,0,0,0,50,0,0,0,0,9,43,0,0,18,52] >;

C10×D6⋊C4 in GAP, Magma, Sage, TeX

C_{10}\times D_6\rtimes C_4
% in TeX

G:=Group("C10xD6:C4");
// GroupNames label

G:=SmallGroup(480,806);
// by ID

G=gap.SmallGroup(480,806);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1766,226,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^6=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

׿
×
𝔽