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## G = Dic10.26D6order 480 = 25·3·5

### 9th non-split extension by Dic10 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic10.26D6
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — S3×Dic10 — Dic10.26D6
 Lower central C15 — C30 — C60 — Dic10.26D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Dic10.26D6
G = < a,b,c,d | a20=d2=1, b2=c6=a10, bab-1=a-1, cac-1=dad=a11, cbc-1=dbd=a5b, dcd=a10c5 >

Subgroups: 556 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3 [×2], C6, C8 [×2], C2×C4 [×3], D4 [×2], Q8, Q8 [×3], C10, C10 [×2], Dic3, Dic3, C12, C12 [×2], D6, D6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×2], C20, C20 [×2], C2×C10 [×2], C3⋊C8, C24, Dic6 [×2], C4×S3, C4×S3 [×2], D12, D12, C3×Q8, C3×Q8, C5×S3 [×2], C30, C8.C22, C52C8, C52C8, Dic10, Dic10 [×2], C2×Dic5, C2×C20 [×2], C5×D4 [×2], C5×Q8, C8⋊S3, C24⋊C2, Q82S3, C3⋊Q16, C3×Q16, S3×Q8, Q83S3, C5×Dic3, C3×Dic5, Dic15, C60, C60, S3×C10, S3×C10, C4.Dic5, D4.D5 [×2], C5⋊Q16, C5⋊Q16, C2×Dic10, C5×C4○D4, Q16⋊S3, C3×C52C8, C153C8, S3×Dic5, C15⋊Q8, C3×Dic10, S3×C20, S3×C20, C5×D12, C5×D12, Dic30, Q8×C15, D4.9D10, D6.Dic5, C20.D6, D12.D5, C3×C5⋊Q16, C157Q16, S3×Dic10, C5×Q83S3, Dic10.26D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8.C22, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, C2×C5⋊D4, Q16⋊S3, C2×S3×D5, D4.9D10, S3×C5⋊D4, Dic10.26D6

Smallest permutation representation of Dic10.26D6
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 27 11 37)(2 26 12 36)(3 25 13 35)(4 24 14 34)(5 23 15 33)(6 22 16 32)(7 21 17 31)(8 40 18 30)(9 39 19 29)(10 38 20 28)(41 144 51 154)(42 143 52 153)(43 142 53 152)(44 141 54 151)(45 160 55 150)(46 159 56 149)(47 158 57 148)(48 157 58 147)(49 156 59 146)(50 155 60 145)(61 178 71 168)(62 177 72 167)(63 176 73 166)(64 175 74 165)(65 174 75 164)(66 173 76 163)(67 172 77 162)(68 171 78 161)(69 170 79 180)(70 169 80 179)(81 125 91 135)(82 124 92 134)(83 123 93 133)(84 122 94 132)(85 121 95 131)(86 140 96 130)(87 139 97 129)(88 138 98 128)(89 137 99 127)(90 136 100 126)(101 188 111 198)(102 187 112 197)(103 186 113 196)(104 185 114 195)(105 184 115 194)(106 183 116 193)(107 182 117 192)(108 181 118 191)(109 200 119 190)(110 199 120 189)(201 228 211 238)(202 227 212 237)(203 226 213 236)(204 225 214 235)(205 224 215 234)(206 223 216 233)(207 222 217 232)(208 221 218 231)(209 240 219 230)(210 239 220 229)
(1 239 86 194 80 49 11 229 96 184 70 59)(2 230 87 185 61 60 12 240 97 195 71 50)(3 221 88 196 62 51 13 231 98 186 72 41)(4 232 89 187 63 42 14 222 99 197 73 52)(5 223 90 198 64 53 15 233 100 188 74 43)(6 234 91 189 65 44 16 224 81 199 75 54)(7 225 92 200 66 55 17 235 82 190 76 45)(8 236 93 191 67 46 18 226 83 181 77 56)(9 227 94 182 68 57 19 237 84 192 78 47)(10 238 95 193 69 48 20 228 85 183 79 58)(21 209 134 114 173 145 31 219 124 104 163 155)(22 220 135 105 174 156 32 210 125 115 164 146)(23 211 136 116 175 147 33 201 126 106 165 157)(24 202 137 107 176 158 34 212 127 117 166 148)(25 213 138 118 177 149 35 203 128 108 167 159)(26 204 139 109 178 160 36 214 129 119 168 150)(27 215 140 120 179 151 37 205 130 110 169 141)(28 206 121 111 180 142 38 216 131 101 170 152)(29 217 122 102 161 153 39 207 132 112 171 143)(30 208 123 113 162 144 40 218 133 103 172 154)
(1 49)(2 60)(3 51)(4 42)(5 53)(6 44)(7 55)(8 46)(9 57)(10 48)(11 59)(12 50)(13 41)(14 52)(15 43)(16 54)(17 45)(18 56)(19 47)(20 58)(21 145)(22 156)(23 147)(24 158)(25 149)(26 160)(27 151)(28 142)(29 153)(30 144)(31 155)(32 146)(33 157)(34 148)(35 159)(36 150)(37 141)(38 152)(39 143)(40 154)(61 230)(62 221)(63 232)(64 223)(65 234)(66 225)(67 236)(68 227)(69 238)(70 229)(71 240)(72 231)(73 222)(74 233)(75 224)(76 235)(77 226)(78 237)(79 228)(80 239)(81 199)(82 190)(83 181)(84 192)(85 183)(86 194)(87 185)(88 196)(89 187)(90 198)(91 189)(92 200)(93 191)(94 182)(95 193)(96 184)(97 195)(98 186)(99 197)(100 188)(101 131)(102 122)(103 133)(104 124)(105 135)(106 126)(107 137)(108 128)(109 139)(110 130)(111 121)(112 132)(113 123)(114 134)(115 125)(116 136)(117 127)(118 138)(119 129)(120 140)(161 217)(162 208)(163 219)(164 210)(165 201)(166 212)(167 203)(168 214)(169 205)(170 216)(171 207)(172 218)(173 209)(174 220)(175 211)(176 202)(177 213)(178 204)(179 215)(180 206)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 5A 5B 6 8A 8B 10A 10B 10C ··· 10H 12A 12B 12C 15A 15B 20A ··· 20F 20G 20H 20I 20J 24A 24B 30A 30B 60A ··· 60F order 1 2 2 2 3 4 4 4 4 4 5 5 6 8 8 10 10 10 ··· 10 12 12 12 15 15 20 ··· 20 20 20 20 20 24 24 30 30 60 ··· 60 size 1 1 6 12 2 2 4 6 20 60 2 2 2 20 60 2 2 12 ··· 12 4 8 40 4 4 4 ··· 4 6 6 6 6 20 20 4 4 8 ··· 8

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + - + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D10 C5⋊D4 C5⋊D4 C8.C22 S3×D4 S3×D5 Q16⋊S3 C2×S3×D5 D4.9D10 S3×C5⋊D4 Dic10.26D6 kernel Dic10.26D6 D6.Dic5 C20.D6 D12.D5 C3×C5⋊Q16 C15⋊7Q16 S3×Dic10 C5×Q8⋊3S3 C5⋊Q16 C5×Dic3 S3×C10 Q8⋊3S3 C5⋊2C8 Dic10 C5×Q8 C4×S3 D12 C3×Q8 Dic3 D6 C15 C10 Q8 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 4 4 1 1 2 2 2 4 4 2

Matrix representation of Dic10.26D6 in GL6(𝔽241)

 189 1 0 0 0 0 240 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 0 0 0 0 0 0 240 0 0
,
 64 16 0 0 0 0 211 177 0 0 0 0 0 0 163 0 120 0 0 0 0 163 0 120 0 0 120 0 78 0 0 0 0 120 0 78
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 93 148 46 195 0 0 93 186 46 92 0 0 46 195 148 93 0 0 46 92 148 55
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 148 93 195 46 0 0 186 93 92 46 0 0 195 46 93 148 0 0 92 46 55 148

G:=sub<GL(6,GF(241))| [189,240,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[64,211,0,0,0,0,16,177,0,0,0,0,0,0,163,0,120,0,0,0,0,163,0,120,0,0,120,0,78,0,0,0,0,120,0,78],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,93,93,46,46,0,0,148,186,195,92,0,0,46,46,148,148,0,0,195,92,93,55],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,148,186,195,92,0,0,93,93,46,46,0,0,195,92,93,55,0,0,46,46,148,148] >;

Dic10.26D6 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}._{26}D_6
% in TeX

G:=Group("Dic10.26D6");
// GroupNames label

G:=SmallGroup(480,586);
// by ID

G=gap.SmallGroup(480,586);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,219,100,675,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=d^2=1,b^2=c^6=a^10,b*a*b^-1=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=d*b*d=a^5*b,d*c*d=a^10*c^5>;
// generators/relations

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