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## G = C10×C6.D4order 480 = 25·3·5

### Direct product of C10 and C6.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C10×C6.D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — Dic3×C2×C10 — C10×C6.D4
 Lower central C3 — C6 — C10×C6.D4
 Upper central C1 — C22×C10 — C23×C10

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 [×6], C2 [×4], C3, C4 [×4], C22, C22 [×10], C22 [×12], C5, C6, C6 [×6], C6 [×4], C2×C4 [×8], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], Dic3 [×4], C2×C6, C2×C6 [×10], C2×C6 [×12], C15, C22⋊C4 [×4], C22×C4 [×2], C24, C20 [×4], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×Dic3 [×4], C2×Dic3 [×4], C22×C6, C22×C6 [×6], C22×C6 [×4], C30, C30 [×6], C30 [×4], C2×C22⋊C4, C2×C20 [×8], C22×C10, C22×C10 [×6], C22×C10 [×4], C6.D4 [×4], C22×Dic3 [×2], C23×C6, C5×Dic3 [×4], C2×C30, C2×C30 [×10], C2×C30 [×12], C5×C22⋊C4 [×4], C22×C20 [×2], C23×C10, C2×C6.D4, C10×Dic3 [×4], C10×Dic3 [×4], C22×C30, C22×C30 [×6], C22×C30 [×4], C10×C22⋊C4, C5×C6.D4 [×4], Dic3×C2×C10 [×2], C23×C30, C10×C6.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], D4 [×4], C23, C10 [×7], Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C5×S3, C2×C22⋊C4, C2×C20 [×6], C5×D4 [×4], C22×C10, C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C5×Dic3 [×4], S3×C10 [×3], C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], C2×C6.D4, C10×Dic3 [×6], C5×C3⋊D4 [×4], S3×C2×C10, C10×C22⋊C4, C5×C6.D4 [×4], Dic3×C2×C10, C10×C3⋊D4 [×2], C10×C6.D4

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

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

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

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

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

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