Copied to
clipboard

## G = Dic13.D4order 416 = 25·13

### 1st non-split extension by Dic13 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — Dic13.D4
 Chief series C1 — C13 — C26 — Dic13 — C2×Dic13 — C13⋊M4(2) — Dic13.D4
 Lower central C13 — C26 — C2×C26 — Dic13.D4
 Upper central C1 — C2 — C22 — C2×C4

Generators and relations for Dic13.D4
G = < a,b,c,d | a26=1, b2=c4=a13, d2=b, bab-1=a-1, cac-1=dad-1=a5, cbc-1=a13b, bd=db, dcd-1=bc3 >

Smallest permutation representation of Dic13.D4
On 208 points
Generators in S208
(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)
(1 40 14 27)(2 39 15 52)(3 38 16 51)(4 37 17 50)(5 36 18 49)(6 35 19 48)(7 34 20 47)(8 33 21 46)(9 32 22 45)(10 31 23 44)(11 30 24 43)(12 29 25 42)(13 28 26 41)(53 84 66 97)(54 83 67 96)(55 82 68 95)(56 81 69 94)(57 80 70 93)(58 79 71 92)(59 104 72 91)(60 103 73 90)(61 102 74 89)(62 101 75 88)(63 100 76 87)(64 99 77 86)(65 98 78 85)(105 148 118 135)(106 147 119 134)(107 146 120 133)(108 145 121 132)(109 144 122 131)(110 143 123 156)(111 142 124 155)(112 141 125 154)(113 140 126 153)(114 139 127 152)(115 138 128 151)(116 137 129 150)(117 136 130 149)(157 197 170 184)(158 196 171 183)(159 195 172 208)(160 194 173 207)(161 193 174 206)(162 192 175 205)(163 191 176 204)(164 190 177 203)(165 189 178 202)(166 188 179 201)(167 187 180 200)(168 186 181 199)(169 185 182 198)
(1 204 27 163 14 191 40 176)(2 199 52 168 15 186 39 181)(3 194 51 173 16 207 38 160)(4 189 50 178 17 202 37 165)(5 184 49 157 18 197 36 170)(6 205 48 162 19 192 35 175)(7 200 47 167 20 187 34 180)(8 195 46 172 21 208 33 159)(9 190 45 177 22 203 32 164)(10 185 44 182 23 198 31 169)(11 206 43 161 24 193 30 174)(12 201 42 166 25 188 29 179)(13 196 41 171 26 183 28 158)(53 150 97 116 66 137 84 129)(54 145 96 121 67 132 83 108)(55 140 95 126 68 153 82 113)(56 135 94 105 69 148 81 118)(57 156 93 110 70 143 80 123)(58 151 92 115 71 138 79 128)(59 146 91 120 72 133 104 107)(60 141 90 125 73 154 103 112)(61 136 89 130 74 149 102 117)(62 131 88 109 75 144 101 122)(63 152 87 114 76 139 100 127)(64 147 86 119 77 134 99 106)(65 142 85 124 78 155 98 111)
(1 138 40 128 14 151 27 115)(2 133 39 107 15 146 52 120)(3 154 38 112 16 141 51 125)(4 149 37 117 17 136 50 130)(5 144 36 122 18 131 49 109)(6 139 35 127 19 152 48 114)(7 134 34 106 20 147 47 119)(8 155 33 111 21 142 46 124)(9 150 32 116 22 137 45 129)(10 145 31 121 23 132 44 108)(11 140 30 126 24 153 43 113)(12 135 29 105 25 148 42 118)(13 156 28 110 26 143 41 123)(53 203 84 164 66 190 97 177)(54 198 83 169 67 185 96 182)(55 193 82 174 68 206 95 161)(56 188 81 179 69 201 94 166)(57 183 80 158 70 196 93 171)(58 204 79 163 71 191 92 176)(59 199 104 168 72 186 91 181)(60 194 103 173 73 207 90 160)(61 189 102 178 74 202 89 165)(62 184 101 157 75 197 88 170)(63 205 100 162 76 192 87 175)(64 200 99 167 77 187 86 180)(65 195 98 172 78 208 85 159)

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

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

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

35 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 8A 8B 8C 8D 13A 13B 13C 26A ··· 26I 52A ··· 52L order 1 2 2 4 4 4 4 8 8 8 8 13 13 13 26 ··· 26 52 ··· 52 size 1 1 2 4 26 26 52 52 52 52 52 4 4 4 4 ··· 4 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 2 4 4 4 4 4 type + + + + - + + + - image C1 C2 C2 C4 C4 D4 C4.10D4 C13⋊C4 C2×C13⋊C4 D13.D4 Dic13.D4 kernel Dic13.D4 C13⋊M4(2) C2×Dic26 C2×Dic13 C2×C52 Dic13 C13 C2×C4 C22 C2 C1 # reps 1 2 1 2 2 2 1 3 3 6 12

Matrix representation of Dic13.D4 in GL8(𝔽313)

 52 1 0 0 0 0 0 0 11 241 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 0 312 261 0 0 0 0 0 0 0 0 312 0 0 0 0 0 0 0 0 312 0 0 0 0 0 0 0 0 312 0 0 0 0 0 0 0 0 312
,
 135 191 0 0 0 0 0 0 252 178 0 0 0 0 0 0 287 203 276 300 0 0 0 0 272 14 33 37 0 0 0 0 0 0 0 0 78 249 0 0 0 0 0 0 188 235 0 0 0 0 0 0 235 273 176 54 0 0 0 0 112 212 290 137
,
 125 127 154 11 0 0 0 0 295 205 150 133 0 0 0 0 167 163 41 40 0 0 0 0 303 100 127 255 0 0 0 0 0 0 0 0 276 0 48 116 0 0 0 0 113 0 51 215 0 0 0 0 245 15 30 229 0 0 0 0 272 221 154 7
,
 125 127 154 11 0 0 0 0 295 205 150 133 0 0 0 0 167 163 41 40 0 0 0 0 303 100 127 255 0 0 0 0 0 0 0 0 221 0 22 0 0 0 0 0 242 0 25 1 0 0 0 0 131 54 92 0 0 0 0 0 84 137 201 0

G:=sub<GL(8,GF(313))| [52,11,0,2,0,0,0,0,1,241,0,0,0,0,0,0,0,0,0,312,0,0,0,0,0,0,1,261,0,0,0,0,0,0,0,0,312,0,0,0,0,0,0,0,0,312,0,0,0,0,0,0,0,0,312,0,0,0,0,0,0,0,0,312],[135,252,287,272,0,0,0,0,191,178,203,14,0,0,0,0,0,0,276,33,0,0,0,0,0,0,300,37,0,0,0,0,0,0,0,0,78,188,235,112,0,0,0,0,249,235,273,212,0,0,0,0,0,0,176,290,0,0,0,0,0,0,54,137],[125,295,167,303,0,0,0,0,127,205,163,100,0,0,0,0,154,150,41,127,0,0,0,0,11,133,40,255,0,0,0,0,0,0,0,0,276,113,245,272,0,0,0,0,0,0,15,221,0,0,0,0,48,51,30,154,0,0,0,0,116,215,229,7],[125,295,167,303,0,0,0,0,127,205,163,100,0,0,0,0,154,150,41,127,0,0,0,0,11,133,40,255,0,0,0,0,0,0,0,0,221,242,131,84,0,0,0,0,0,0,54,137,0,0,0,0,22,25,92,201,0,0,0,0,0,1,0,0] >;

Dic13.D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{13}.D_4
% in TeX

G:=Group("Dic13.D4");
// GroupNames label

G:=SmallGroup(416,80);
// by ID

G=gap.SmallGroup(416,80);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,103,188,86,579,9221,3473]);
// Polycyclic

G:=Group<a,b,c,d|a^26=1,b^2=c^4=a^13,d^2=b,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^5,c*b*c^-1=a^13*b,b*d=d*b,d*c*d^-1=b*c^3>;
// generators/relations

Export

׿
×
𝔽