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## G = SD16×C26order 416 = 25·13

### Direct product of C26 and SD16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — SD16×C26
 Chief series C1 — C2 — C4 — C52 — Q8×C13 — C13×SD16 — SD16×C26
 Lower central C1 — C2 — C4 — SD16×C26
 Upper central C1 — C2×C26 — C2×C52 — SD16×C26

Generators and relations for SD16×C26
G = < a,b,c | a26=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Subgroups: 108 in 68 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C8 [×2], C2×C4, C2×C4, D4 [×2], D4, Q8 [×2], Q8, C23, C13, C2×C8, SD16 [×4], C2×D4, C2×Q8, C26, C26 [×2], C26 [×2], C2×SD16, C52 [×2], C52 [×2], C2×C26, C2×C26 [×4], C104 [×2], C2×C52, C2×C52, D4×C13 [×2], D4×C13, Q8×C13 [×2], Q8×C13, C22×C26, C2×C104, C13×SD16 [×4], D4×C26, Q8×C26, SD16×C26
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C13, SD16 [×2], C2×D4, C26 [×7], C2×SD16, C2×C26 [×7], D4×C13 [×2], C22×C26, C13×SD16 [×2], D4×C26, SD16×C26

Smallest permutation representation of SD16×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 41 165 59 121 81 203 135)(2 42 166 60 122 82 204 136)(3 43 167 61 123 83 205 137)(4 44 168 62 124 84 206 138)(5 45 169 63 125 85 207 139)(6 46 170 64 126 86 208 140)(7 47 171 65 127 87 183 141)(8 48 172 66 128 88 184 142)(9 49 173 67 129 89 185 143)(10 50 174 68 130 90 186 144)(11 51 175 69 105 91 187 145)(12 52 176 70 106 92 188 146)(13 27 177 71 107 93 189 147)(14 28 178 72 108 94 190 148)(15 29 179 73 109 95 191 149)(16 30 180 74 110 96 192 150)(17 31 181 75 111 97 193 151)(18 32 182 76 112 98 194 152)(19 33 157 77 113 99 195 153)(20 34 158 78 114 100 196 154)(21 35 159 53 115 101 197 155)(22 36 160 54 116 102 198 156)(23 37 161 55 117 103 199 131)(24 38 162 56 118 104 200 132)(25 39 163 57 119 79 201 133)(26 40 164 58 120 80 202 134)
(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(79 133)(80 134)(81 135)(82 136)(83 137)(84 138)(85 139)(86 140)(87 141)(88 142)(89 143)(90 144)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 151)(98 152)(99 153)(100 154)(101 155)(102 156)(103 131)(104 132)(157 195)(158 196)(159 197)(160 198)(161 199)(162 200)(163 201)(164 202)(165 203)(166 204)(167 205)(168 206)(169 207)(170 208)(171 183)(172 184)(173 185)(174 186)(175 187)(176 188)(177 189)(178 190)(179 191)(180 192)(181 193)(182 194)

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

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

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

182 conjugacy classes

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

182 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C13 C26 C26 C26 C26 D4 D4 SD16 D4×C13 D4×C13 C13×SD16 kernel SD16×C26 C2×C104 C13×SD16 D4×C26 Q8×C26 C2×SD16 C2×C8 SD16 C2×D4 C2×Q8 C52 C2×C26 C26 C4 C22 C2 # reps 1 1 4 1 1 12 12 48 12 12 1 1 4 12 12 48

Matrix representation of SD16×C26 in GL3(𝔽313) generated by

 312 0 0 0 48 0 0 0 48
,
 1 0 0 0 248 65 0 248 248
,
 312 0 0 0 1 0 0 0 312
G:=sub<GL(3,GF(313))| [312,0,0,0,48,0,0,0,48],[1,0,0,0,248,248,0,65,248],[312,0,0,0,1,0,0,0,312] >;

SD16×C26 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times C_{26}
% in TeX

G:=Group("SD16xC26");
// GroupNames label

G:=SmallGroup(416,194);
// by ID

G=gap.SmallGroup(416,194);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1248,1273,9364,4690,88]);
// 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^3>;
// generators/relations

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