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## G = C2×D13⋊C8order 416 = 25·13

### Direct product of C2 and D13⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C2×D13⋊C8
 Chief series C1 — C13 — C26 — Dic13 — C13⋊C8 — C2×C13⋊C8 — C2×D13⋊C8
 Lower central C13 — C2×D13⋊C8
 Upper central C1 — C2×C4

Generators and relations for C2×D13⋊C8
G = < a,b,c,d | a2=b13=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b4c >

Subgroups: 436 in 76 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C22, C22 [×6], C8 [×4], C2×C4, C2×C4 [×5], C23, C13, C2×C8 [×6], C22×C4, D13 [×4], C26, C26 [×2], C22×C8, Dic13 [×2], C52 [×2], D26 [×6], C2×C26, C13⋊C8 [×4], C4×D13 [×4], C2×Dic13, C2×C52, C22×D13, D13⋊C8 [×4], C2×C13⋊C8 [×2], C2×C4×D13, C2×D13⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, C22×C8, C13⋊C4, C2×C13⋊C4 [×3], D13⋊C8 [×2], C22×C13⋊C4, C2×D13⋊C8

Smallest permutation representation of C2×D13⋊C8
On 208 points
Generators in S208
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 89)(28 90)(29 91)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 92)(49 93)(50 94)(51 95)(52 96)(105 159)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)(113 167)(114 168)(115 169)(116 157)(117 158)(118 171)(119 172)(120 173)(121 174)(122 175)(123 176)(124 177)(125 178)(126 179)(127 180)(128 181)(129 182)(130 170)(131 192)(132 193)(133 194)(134 195)(135 183)(136 184)(137 185)(138 186)(139 187)(140 188)(141 189)(142 190)(143 191)(144 201)(145 202)(146 203)(147 204)(148 205)(149 206)(150 207)(151 208)(152 196)(153 197)(154 198)(155 199)(156 200)
(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 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 71)(22 70)(23 69)(24 68)(25 67)(26 66)(27 81)(28 80)(29 79)(30 91)(31 90)(32 89)(33 88)(34 87)(35 86)(36 85)(37 84)(38 83)(39 82)(40 97)(41 96)(42 95)(43 94)(44 93)(45 92)(46 104)(47 103)(48 102)(49 101)(50 100)(51 99)(52 98)(105 167)(106 166)(107 165)(108 164)(109 163)(110 162)(111 161)(112 160)(113 159)(114 158)(115 157)(116 169)(117 168)(118 180)(119 179)(120 178)(121 177)(122 176)(123 175)(124 174)(125 173)(126 172)(127 171)(128 170)(129 182)(130 181)(131 192)(132 191)(133 190)(134 189)(135 188)(136 187)(137 186)(138 185)(139 184)(140 183)(141 195)(142 194)(143 193)(144 207)(145 206)(146 205)(147 204)(148 203)(149 202)(150 201)(151 200)(152 199)(153 198)(154 197)(155 196)(156 208)
(1 154 47 123 14 138 30 116)(2 149 46 128 15 133 29 108)(3 144 45 120 16 141 28 113)(4 152 44 125 17 136 27 105)(5 147 43 130 18 131 39 110)(6 155 42 122 19 139 38 115)(7 150 41 127 20 134 37 107)(8 145 40 119 21 142 36 112)(9 153 52 124 22 137 35 117)(10 148 51 129 23 132 34 109)(11 156 50 121 24 140 33 114)(12 151 49 126 25 135 32 106)(13 146 48 118 26 143 31 111)(53 198 104 176 66 186 79 157)(54 206 103 181 67 194 91 162)(55 201 102 173 68 189 90 167)(56 196 101 178 69 184 89 159)(57 204 100 170 70 192 88 164)(58 199 99 175 71 187 87 169)(59 207 98 180 72 195 86 161)(60 202 97 172 73 190 85 166)(61 197 96 177 74 185 84 158)(62 205 95 182 75 193 83 163)(63 200 94 174 76 188 82 168)(64 208 93 179 77 183 81 160)(65 203 92 171 78 191 80 165)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 8A ··· 8P 13A 13B 13C 26A ··· 26I 52A ··· 52L order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 ··· 8 13 13 13 26 ··· 26 52 ··· 52 size 1 1 1 1 13 13 13 13 1 1 1 1 13 13 13 13 13 ··· 13 4 4 4 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 C8 C13⋊C4 C2×C13⋊C4 C2×C13⋊C4 D13⋊C8 kernel C2×D13⋊C8 D13⋊C8 C2×C13⋊C8 C2×C4×D13 C4×D13 C2×C52 C22×D13 D26 C2×C4 C4 C22 C2 # reps 1 4 2 1 4 2 2 16 3 6 3 12

Matrix representation of C2×D13⋊C8 in GL5(𝔽313)

 312 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 312 312 312 70 0 1 0 0 31 0 0 1 0 282 0 0 0 1 242
,
 1 0 0 0 0 0 72 30 62 61 0 30 312 39 39 0 210 0 272 273 0 241 1 211 283
,
 1 0 0 0 0 0 284 225 293 19 0 70 88 271 271 0 124 143 44 5 0 172 0 100 210

G:=sub<GL(5,GF(313))| [312,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,312,1,0,0,0,312,0,1,0,0,312,0,0,1,0,70,31,282,242],[1,0,0,0,0,0,72,30,210,241,0,30,312,0,1,0,62,39,272,211,0,61,39,273,283],[1,0,0,0,0,0,284,70,124,172,0,225,88,143,0,0,293,271,44,100,0,19,271,5,210] >;

C2×D13⋊C8 in GAP, Magma, Sage, TeX

C_2\times D_{13}\rtimes C_8
% in TeX

G:=Group("C2xD13:C8");
// GroupNames label

G:=SmallGroup(416,199);
// by ID

G=gap.SmallGroup(416,199);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,86,69,9221,1751]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^13=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^4*c>;
// generators/relations

׿
×
𝔽