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## G = D4.Dic13order 416 = 25·13

### The non-split extension by D4 of Dic13 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D4.Dic13
 Chief series C1 — C13 — C26 — C52 — C13⋊2C8 — C2×C13⋊2C8 — D4.Dic13
 Lower central C13 — C26 — D4.Dic13
 Upper central C1 — C4 — C4○D4

Generators and relations for D4.Dic13
G = < a,b,c,d | a4=1, b2=c26=a2, d2=c13, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c25 >

Subgroups: 200 in 62 conjugacy classes, 45 normal (12 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C8 [×4], C2×C4 [×3], D4 [×3], Q8, C13, C2×C8 [×3], M4(2) [×3], C4○D4, C26, C26 [×3], C8○D4, C52, C52 [×3], C2×C26 [×3], C132C8, C132C8 [×3], C2×C52 [×3], D4×C13 [×3], Q8×C13, C2×C132C8 [×3], C52.4C4 [×3], C13×C4○D4, D4.Dic13
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, D13, C8○D4, Dic13 [×4], D26 [×3], C2×Dic13 [×6], C22×D13, C22×Dic13, D4.Dic13

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

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E ··· 8J 13A ··· 13F 26A ··· 26F 26G ··· 26X 52A ··· 52L 52M ··· 52AD order 1 2 2 2 2 4 4 4 4 4 8 8 8 8 8 ··· 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 2 2 2 1 1 2 2 2 13 13 13 13 26 ··· 26 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + - - image C1 C2 C2 C2 C4 C4 D13 C8○D4 D26 Dic13 Dic13 D4.Dic13 kernel D4.Dic13 C2×C13⋊2C8 C52.4C4 C13×C4○D4 D4×C13 Q8×C13 C4○D4 C13 C2×C4 D4 Q8 C1 # reps 1 3 3 1 6 2 6 4 18 18 6 12

Matrix representation of D4.Dic13 in GL4(𝔽313) generated by

 312 0 0 0 0 312 0 0 0 0 288 197 0 0 0 25
,
 1 0 0 0 0 1 0 0 0 0 288 0 0 0 286 25
,
 0 1 0 0 312 24 0 0 0 0 288 0 0 0 0 288
,
 104 252 0 0 244 209 0 0 0 0 188 0 0 0 0 188
G:=sub<GL(4,GF(313))| [312,0,0,0,0,312,0,0,0,0,288,0,0,0,197,25],[1,0,0,0,0,1,0,0,0,0,288,286,0,0,0,25],[0,312,0,0,1,24,0,0,0,0,288,0,0,0,0,288],[104,244,0,0,252,209,0,0,0,0,188,0,0,0,0,188] >;

D4.Dic13 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(416,169);
// by ID

G=gap.SmallGroup(416,169);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,188,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^4=1,b^2=c^26=a^2,d^2=c^13,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^25>;
// generators/relations

׿
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