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## G = D8×C26order 416 = 25·13

### Direct product of C26 and D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — D8×C26
 Chief series C1 — C2 — C4 — C52 — D4×C13 — C13×D8 — D8×C26
 Lower central C1 — C2 — C4 — D8×C26
 Upper central C1 — C2×C26 — C2×C52 — D8×C26

Generators and relations for D8×C26
G = < a,b,c | a26=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 140 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C22, C22 [×8], C8 [×2], C2×C4, D4 [×4], D4 [×2], C23 [×2], C13, C2×C8, D8 [×4], C2×D4 [×2], C26, C26 [×2], C26 [×4], C2×D8, C52 [×2], C2×C26, C2×C26 [×8], C104 [×2], C2×C52, D4×C13 [×4], D4×C13 [×2], C22×C26 [×2], C2×C104, C13×D8 [×4], D4×C26 [×2], D8×C26
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C13, D8 [×2], C2×D4, C26 [×7], C2×D8, C2×C26 [×7], D4×C13 [×2], C22×C26, C13×D8 [×2], D4×C26, D8×C26

Smallest permutation representation of D8×C26
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)
(1 192 86 106 134 41 65 171)(2 193 87 107 135 42 66 172)(3 194 88 108 136 43 67 173)(4 195 89 109 137 44 68 174)(5 196 90 110 138 45 69 175)(6 197 91 111 139 46 70 176)(7 198 92 112 140 47 71 177)(8 199 93 113 141 48 72 178)(9 200 94 114 142 49 73 179)(10 201 95 115 143 50 74 180)(11 202 96 116 144 51 75 181)(12 203 97 117 145 52 76 182)(13 204 98 118 146 27 77 157)(14 205 99 119 147 28 78 158)(15 206 100 120 148 29 53 159)(16 207 101 121 149 30 54 160)(17 208 102 122 150 31 55 161)(18 183 103 123 151 32 56 162)(19 184 104 124 152 33 57 163)(20 185 79 125 153 34 58 164)(21 186 80 126 154 35 59 165)(22 187 81 127 155 36 60 166)(23 188 82 128 156 37 61 167)(24 189 83 129 131 38 62 168)(25 190 84 130 132 39 63 169)(26 191 85 105 133 40 64 170)
(1 147)(2 148)(3 149)(4 150)(5 151)(6 152)(7 153)(8 154)(9 155)(10 156)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 170)(28 171)(29 172)(30 173)(31 174)(32 175)(33 176)(34 177)(35 178)(36 179)(37 180)(38 181)(39 182)(40 157)(41 158)(42 159)(43 160)(44 161)(45 162)(46 163)(47 164)(48 165)(49 166)(50 167)(51 168)(52 169)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)(105 204)(106 205)(107 206)(108 207)(109 208)(110 183)(111 184)(112 185)(113 186)(114 187)(115 188)(116 189)(117 190)(118 191)(119 192)(120 193)(121 194)(122 195)(123 196)(124 197)(125 198)(126 199)(127 200)(128 201)(129 202)(130 203)

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

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

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

182 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 8A 8B 8C 8D 13A ··· 13L 26A ··· 26AJ 26AK ··· 26CF 52A ··· 52X 104A ··· 104AV order 1 2 2 2 2 2 2 2 4 4 8 8 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 104 ··· 104 size 1 1 1 1 4 4 4 4 2 2 2 2 2 2 1 ··· 1 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2

182 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C13 C26 C26 C26 D4 D4 D8 D4×C13 D4×C13 C13×D8 kernel D8×C26 C2×C104 C13×D8 D4×C26 C2×D8 C2×C8 D8 C2×D4 C52 C2×C26 C26 C4 C22 C2 # reps 1 1 4 2 12 12 48 24 1 1 4 12 12 48

Matrix representation of D8×C26 in GL3(𝔽313) generated by

 312 0 0 0 64 0 0 0 64
,
 1 0 0 0 0 120 0 253 120
,
 1 0 0 0 1 0 0 1 312
G:=sub<GL(3,GF(313))| [312,0,0,0,64,0,0,0,64],[1,0,0,0,0,253,0,120,120],[1,0,0,0,1,1,0,0,312] >;

D8×C26 in GAP, Magma, Sage, TeX

D_8\times C_{26}
% in TeX

G:=Group("D8xC26");
// GroupNames label

G:=SmallGroup(416,193);
// by ID

G=gap.SmallGroup(416,193);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,9364,4690,88]);
// Polycyclic

G:=Group<a,b,c|a^26=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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