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## G = C13×C8○D4order 416 = 25·13

### Direct product of C13 and C8○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C13×C8○D4
 Chief series C1 — C2 — C4 — C52 — C104 — C2×C104 — C13×C8○D4
 Lower central C1 — C2 — C13×C8○D4
 Upper central C1 — C104 — C13×C8○D4

Generators and relations for C13×C8○D4
G = < a,b,c,d | a13=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 68 in 62 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C8, C8 [×3], 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], C104, C104 [×3], C2×C52 [×3], D4×C13 [×3], Q8×C13, C2×C104 [×3], C13×M4(2) [×3], C13×C4○D4, C13×C8○D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C13, C22×C4, C26 [×7], C8○D4, C52 [×4], C2×C26 [×7], C2×C52 [×6], C22×C26, C22×C52, C13×C8○D4

Smallest permutation representation of C13×C8○D4
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)
(1 187 83 161 57 181 77 146)(2 188 84 162 58 182 78 147)(3 189 85 163 59 170 66 148)(4 190 86 164 60 171 67 149)(5 191 87 165 61 172 68 150)(6 192 88 166 62 173 69 151)(7 193 89 167 63 174 70 152)(8 194 90 168 64 175 71 153)(9 195 91 169 65 176 72 154)(10 183 79 157 53 177 73 155)(11 184 80 158 54 178 74 156)(12 185 81 159 55 179 75 144)(13 186 82 160 56 180 76 145)(14 113 33 120 202 98 52 139)(15 114 34 121 203 99 40 140)(16 115 35 122 204 100 41 141)(17 116 36 123 205 101 42 142)(18 117 37 124 206 102 43 143)(19 105 38 125 207 103 44 131)(20 106 39 126 208 104 45 132)(21 107 27 127 196 92 46 133)(22 108 28 128 197 93 47 134)(23 109 29 129 198 94 48 135)(24 110 30 130 199 95 49 136)(25 111 31 118 200 96 50 137)(26 112 32 119 201 97 51 138)
(1 94 57 109)(2 95 58 110)(3 96 59 111)(4 97 60 112)(5 98 61 113)(6 99 62 114)(7 100 63 115)(8 101 64 116)(9 102 65 117)(10 103 53 105)(11 104 54 106)(12 92 55 107)(13 93 56 108)(14 150 202 165)(15 151 203 166)(16 152 204 167)(17 153 205 168)(18 154 206 169)(19 155 207 157)(20 156 208 158)(21 144 196 159)(22 145 197 160)(23 146 198 161)(24 147 199 162)(25 148 200 163)(26 149 201 164)(27 185 46 179)(28 186 47 180)(29 187 48 181)(30 188 49 182)(31 189 50 170)(32 190 51 171)(33 191 52 172)(34 192 40 173)(35 193 41 174)(36 194 42 175)(37 195 43 176)(38 183 44 177)(39 184 45 178)(66 118 85 137)(67 119 86 138)(68 120 87 139)(69 121 88 140)(70 122 89 141)(71 123 90 142)(72 124 91 143)(73 125 79 131)(74 126 80 132)(75 127 81 133)(76 128 82 134)(77 129 83 135)(78 130 84 136)
(14 202)(15 203)(16 204)(17 205)(18 206)(19 207)(20 208)(21 196)(22 197)(23 198)(24 199)(25 200)(26 201)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 40)(35 41)(36 42)(37 43)(38 44)(39 45)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 105)(104 106)(118 137)(119 138)(120 139)(121 140)(122 141)(123 142)(124 143)(125 131)(126 132)(127 133)(128 134)(129 135)(130 136)

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

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

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

260 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E ··· 8J 13A ··· 13L 26A ··· 26L 26M ··· 26AV 52A ··· 52X 52Y ··· 52BH 104A ··· 104AV 104AW ··· 104DP 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 ··· 52 104 ··· 104 104 ··· 104 size 1 1 2 2 2 1 1 2 2 2 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

260 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C4 C4 C13 C26 C26 C26 C52 C52 C8○D4 C13×C8○D4 kernel C13×C8○D4 C2×C104 C13×M4(2) C13×C4○D4 D4×C13 Q8×C13 C8○D4 C2×C8 M4(2) C4○D4 D4 Q8 C13 C1 # reps 1 3 3 1 6 2 12 36 36 12 72 24 4 48

Matrix representation of C13×C8○D4 in GL2(𝔽313) generated by

 113 0 0 113
,
 5 0 0 5
,
 265 2 256 48
,
 1 0 48 312
G:=sub<GL(2,GF(313))| [113,0,0,113],[5,0,0,5],[265,256,2,48],[1,48,0,312] >;

C13×C8○D4 in GAP, Magma, Sage, TeX

C_{13}\times C_8\circ D_4
% in TeX

G:=Group("C13xC8oD4");
// GroupNames label

G:=SmallGroup(416,192);
// by ID

G=gap.SmallGroup(416,192);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,624,1916,88]);
// Polycyclic

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

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