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## G = C13×C4○D8order 416 = 25·13

### Direct product of C13 and C4○D8

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C13×C4○D8
G = < a,b,c,d | a13=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 92 in 62 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C13, C2×C8, D8, SD16, Q16, C4○D4, C26, C26, C4○D8, C52, C52, C2×C26, C2×C26, C104, C2×C52, C2×C52, D4×C13, D4×C13, Q8×C13, C2×C104, C13×D8, C13×SD16, C13×Q16, C13×C4○D4, C13×C4○D8
Quotients: C1, C2, C22, D4, C23, C13, C2×D4, C26, C4○D8, C2×C26, D4×C13, C22×C26, D4×C26, C13×C4○D8

Smallest permutation representation of C13×C4○D8
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)
(1 137 53 192)(2 138 54 193)(3 139 55 194)(4 140 56 195)(5 141 57 183)(6 142 58 184)(7 143 59 185)(8 131 60 186)(9 132 61 187)(10 133 62 188)(11 134 63 189)(12 135 64 190)(13 136 65 191)(14 103 181 113)(15 104 182 114)(16 92 170 115)(17 93 171 116)(18 94 172 117)(19 95 173 105)(20 96 174 106)(21 97 175 107)(22 98 176 108)(23 99 177 109)(24 100 178 110)(25 101 179 111)(26 102 180 112)(27 119 82 203)(28 120 83 204)(29 121 84 205)(30 122 85 206)(31 123 86 207)(32 124 87 208)(33 125 88 196)(34 126 89 197)(35 127 90 198)(36 128 91 199)(37 129 79 200)(38 130 80 201)(39 118 81 202)(40 160 147 77)(41 161 148 78)(42 162 149 66)(43 163 150 67)(44 164 151 68)(45 165 152 69)(46 166 153 70)(47 167 154 71)(48 168 155 72)(49 169 156 73)(50 157 144 74)(51 158 145 75)(52 159 146 76)
(1 203 137 27 53 119 192 82)(2 204 138 28 54 120 193 83)(3 205 139 29 55 121 194 84)(4 206 140 30 56 122 195 85)(5 207 141 31 57 123 183 86)(6 208 142 32 58 124 184 87)(7 196 143 33 59 125 185 88)(8 197 131 34 60 126 186 89)(9 198 132 35 61 127 187 90)(10 199 133 36 62 128 188 91)(11 200 134 37 63 129 189 79)(12 201 135 38 64 130 190 80)(13 202 136 39 65 118 191 81)(14 169 113 49 181 73 103 156)(15 157 114 50 182 74 104 144)(16 158 115 51 170 75 92 145)(17 159 116 52 171 76 93 146)(18 160 117 40 172 77 94 147)(19 161 105 41 173 78 95 148)(20 162 106 42 174 66 96 149)(21 163 107 43 175 67 97 150)(22 164 108 44 176 68 98 151)(23 165 109 45 177 69 99 152)(24 166 110 46 178 70 100 153)(25 167 111 47 179 71 101 154)(26 168 112 48 180 72 102 155)
(1 148)(2 149)(3 150)(4 151)(5 152)(6 153)(7 154)(8 155)(9 156)(10 144)(11 145)(12 146)(13 147)(14 90)(15 91)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 173)(28 174)(29 175)(30 176)(31 177)(32 178)(33 179)(34 180)(35 181)(36 182)(37 170)(38 171)(39 172)(40 65)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)(49 61)(50 62)(51 63)(52 64)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 131)(73 132)(74 133)(75 134)(76 135)(77 136)(78 137)(92 200)(93 201)(94 202)(95 203)(96 204)(97 205)(98 206)(99 207)(100 208)(101 196)(102 197)(103 198)(104 199)(105 119)(106 120)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 118)(157 188)(158 189)(159 190)(160 191)(161 192)(162 193)(163 194)(164 195)(165 183)(166 184)(167 185)(168 186)(169 187)

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

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

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

182 conjugacy classes

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

182 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C13 C26 C26 C26 C26 C26 D4 D4 C4○D8 D4×C13 D4×C13 C13×C4○D8 kernel C13×C4○D8 C2×C104 C13×D8 C13×SD16 C13×Q16 C13×C4○D4 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 C52 C2×C26 C13 C4 C22 C1 # reps 1 1 1 2 1 2 12 12 12 24 12 24 1 1 4 12 12 48

Matrix representation of C13×C4○D8 in GL3(𝔽313) generated by

 48 0 0 0 1 0 0 0 1
,
 312 0 0 0 288 0 0 0 288
,
 1 0 0 0 188 0 0 0 5
,
 312 0 0 0 0 5 0 188 0
G:=sub<GL(3,GF(313))| [48,0,0,0,1,0,0,0,1],[312,0,0,0,288,0,0,0,288],[1,0,0,0,188,0,0,0,5],[312,0,0,0,0,188,0,5,0] >;

C13×C4○D8 in GAP, Magma, Sage, TeX

C_{13}\times C_4\circ D_8
% in TeX

G:=Group("C13xC4oD8");
// GroupNames label

G:=SmallGroup(416,196);
// by ID

G=gap.SmallGroup(416,196);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,950,9364,4690,88]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations

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