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## G = Dic26.C4order 416 = 25·13

### The non-split extension by Dic26 of C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — Dic26.C4
 Chief series C1 — C13 — C26 — Dic13 — C13⋊C8 — C2×C13⋊C8 — Dic26.C4
 Lower central C13 — C26 — Dic26.C4
 Upper central C1 — C2 — D4

Generators and relations for Dic26.C4
G = < a,b,c | a52=1, b2=c4=a26, bab-1=a-1, cac-1=a5, bc=cb >

Subgroups: 356 in 62 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×2], C22, C8 [×4], C2×C4 [×3], D4, D4 [×2], Q8, C13, C2×C8 [×3], M4(2) [×3], C4○D4, D13, C26, C26 [×2], C8○D4, Dic13, Dic13 [×2], C52, D26, C2×C26 [×2], C13⋊C8 [×2], C13⋊C8 [×2], Dic26, C4×D13, C2×Dic13 [×2], C13⋊D4 [×2], D4×C13, D13⋊C8, C52.C4, C2×C13⋊C8 [×2], C13⋊M4(2) [×2], D42D13, Dic26.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C8○D4, C13⋊C4, C2×C13⋊C4 [×3], C22×C13⋊C4, Dic26.C4

Smallest permutation representation of Dic26.C4
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)
(1 110 27 136)(2 109 28 135)(3 108 29 134)(4 107 30 133)(5 106 31 132)(6 105 32 131)(7 156 33 130)(8 155 34 129)(9 154 35 128)(10 153 36 127)(11 152 37 126)(12 151 38 125)(13 150 39 124)(14 149 40 123)(15 148 41 122)(16 147 42 121)(17 146 43 120)(18 145 44 119)(19 144 45 118)(20 143 46 117)(21 142 47 116)(22 141 48 115)(23 140 49 114)(24 139 50 113)(25 138 51 112)(26 137 52 111)(53 199 79 173)(54 198 80 172)(55 197 81 171)(56 196 82 170)(57 195 83 169)(58 194 84 168)(59 193 85 167)(60 192 86 166)(61 191 87 165)(62 190 88 164)(63 189 89 163)(64 188 90 162)(65 187 91 161)(66 186 92 160)(67 185 93 159)(68 184 94 158)(69 183 95 157)(70 182 96 208)(71 181 97 207)(72 180 98 206)(73 179 99 205)(74 178 100 204)(75 177 101 203)(76 176 102 202)(77 175 103 201)(78 174 104 200)
(1 195 149 70 27 169 123 96)(2 164 122 75 28 190 148 101)(3 185 147 80 29 159 121 54)(4 206 120 85 30 180 146 59)(5 175 145 90 31 201 119 64)(6 196 118 95 32 170 144 69)(7 165 143 100 33 191 117 74)(8 186 116 53 34 160 142 79)(9 207 141 58 35 181 115 84)(10 176 114 63 36 202 140 89)(11 197 139 68 37 171 113 94)(12 166 112 73 38 192 138 99)(13 187 137 78 39 161 111 104)(14 208 110 83 40 182 136 57)(15 177 135 88 41 203 109 62)(16 198 108 93 42 172 134 67)(17 167 133 98 43 193 107 72)(18 188 106 103 44 162 132 77)(19 157 131 56 45 183 105 82)(20 178 156 61 46 204 130 87)(21 199 129 66 47 173 155 92)(22 168 154 71 48 194 128 97)(23 189 127 76 49 163 153 102)(24 158 152 81 50 184 126 55)(25 179 125 86 51 205 151 60)(26 200 150 91 52 174 124 65)

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

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

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

35 conjugacy classes

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

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 4 4 4 8 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8○D4 C13⋊C4 C2×C13⋊C4 C2×C13⋊C4 Dic26.C4 kernel Dic26.C4 D13⋊C8 C52.C4 C2×C13⋊C8 C13⋊M4(2) D4⋊2D13 Dic26 C13⋊D4 D4×C13 C13 D4 C4 C22 C1 # reps 1 1 1 2 2 1 2 4 2 4 3 3 6 3

Matrix representation of Dic26.C4 in GL6(𝔽313)

 0 312 0 0 0 0 1 0 0 0 0 0 0 0 1 312 0 0 0 0 1 0 312 0 0 0 1 0 0 312 0 0 103 240 73 211
,
 0 288 0 0 0 0 288 0 0 0 0 0 0 0 88 33 304 50 0 0 53 92 289 201 0 0 211 291 171 41 0 0 230 40 244 275
,
 125 0 0 0 0 0 0 125 0 0 0 0 0 0 188 45 53 56 0 0 60 299 163 184 0 0 143 273 269 254 0 0 296 26 303 183

G:=sub<GL(6,GF(313))| [0,1,0,0,0,0,312,0,0,0,0,0,0,0,1,1,1,103,0,0,312,0,0,240,0,0,0,312,0,73,0,0,0,0,312,211],[0,288,0,0,0,0,288,0,0,0,0,0,0,0,88,53,211,230,0,0,33,92,291,40,0,0,304,289,171,244,0,0,50,201,41,275],[125,0,0,0,0,0,0,125,0,0,0,0,0,0,188,60,143,296,0,0,45,299,273,26,0,0,53,163,269,303,0,0,56,184,254,183] >;

Dic26.C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{26}.C_4
% in TeX

G:=Group("Dic26.C4");
// GroupNames label

G:=SmallGroup(416,205);
// by ID

G=gap.SmallGroup(416,205);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,188,69,9221,1751]);
// Polycyclic

G:=Group<a,b,c|a^52=1,b^2=c^4=a^26,b*a*b^-1=a^-1,c*a*c^-1=a^5,b*c=c*b>;
// generators/relations

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