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

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

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

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

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

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