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## G = C13×C42⋊C2order 416 = 25·13

### Direct product of C13 and C42⋊C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C13×C42⋊C2
 Chief series C1 — C2 — C22 — C2×C26 — C2×C52 — C13×C22⋊C4 — C13×C42⋊C2
 Lower central C1 — C2 — C13×C42⋊C2
 Upper central C1 — C2×C52 — C13×C42⋊C2

Generators and relations for C13×C42⋊C2
G = < a,b,c,d | a13=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >

Subgroups: 92 in 76 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C2×C4 [×2], C2×C4 [×8], C23, C13, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C26, C26 [×2], C26 [×2], C42⋊C2, C52 [×4], C52 [×4], C2×C26, C2×C26 [×2], C2×C26 [×2], C2×C52 [×2], C2×C52 [×8], C22×C26, C4×C52 [×2], C13×C22⋊C4 [×2], C13×C4⋊C4 [×2], C22×C52, C13×C42⋊C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C13, C22×C4, C4○D4 [×2], C26 [×7], C42⋊C2, C52 [×4], C2×C26 [×7], C2×C52 [×6], C22×C26, C22×C52, C13×C4○D4 [×2], C13×C42⋊C2

Smallest permutation representation of C13×C42⋊C2
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)
(1 139 204 165)(2 140 205 166)(3 141 206 167)(4 142 207 168)(5 143 208 169)(6 131 196 157)(7 132 197 158)(8 133 198 159)(9 134 199 160)(10 135 200 161)(11 136 201 162)(12 137 202 163)(13 138 203 164)(14 80 110 97)(15 81 111 98)(16 82 112 99)(17 83 113 100)(18 84 114 101)(19 85 115 102)(20 86 116 103)(21 87 117 104)(22 88 105 92)(23 89 106 93)(24 90 107 94)(25 91 108 95)(26 79 109 96)(27 171 153 54)(28 172 154 55)(29 173 155 56)(30 174 156 57)(31 175 144 58)(32 176 145 59)(33 177 146 60)(34 178 147 61)(35 179 148 62)(36 180 149 63)(37 181 150 64)(38 182 151 65)(39 170 152 53)(40 72 120 189)(41 73 121 190)(42 74 122 191)(43 75 123 192)(44 76 124 193)(45 77 125 194)(46 78 126 195)(47 66 127 183)(48 67 128 184)(49 68 129 185)(50 69 130 186)(51 70 118 187)(52 71 119 188)
(1 149 46 21)(2 150 47 22)(3 151 48 23)(4 152 49 24)(5 153 50 25)(6 154 51 26)(7 155 52 14)(8 156 40 15)(9 144 41 16)(10 145 42 17)(11 146 43 18)(12 147 44 19)(13 148 45 20)(27 130 108 208)(28 118 109 196)(29 119 110 197)(30 120 111 198)(31 121 112 199)(32 122 113 200)(33 123 114 201)(34 124 115 202)(35 125 116 203)(36 126 117 204)(37 127 105 205)(38 128 106 206)(39 129 107 207)(53 68 90 142)(54 69 91 143)(55 70 79 131)(56 71 80 132)(57 72 81 133)(58 73 82 134)(59 74 83 135)(60 75 84 136)(61 76 85 137)(62 77 86 138)(63 78 87 139)(64 66 88 140)(65 67 89 141)(92 166 181 183)(93 167 182 184)(94 168 170 185)(95 169 171 186)(96 157 172 187)(97 158 173 188)(98 159 174 189)(99 160 175 190)(100 161 176 191)(101 162 177 192)(102 163 178 193)(103 164 179 194)(104 165 180 195)
(1 204)(2 205)(3 206)(4 207)(5 208)(6 196)(7 197)(8 198)(9 199)(10 200)(11 201)(12 202)(13 203)(14 110)(15 111)(16 112)(17 113)(18 114)(19 115)(20 116)(21 117)(22 105)(23 106)(24 107)(25 108)(26 109)(27 153)(28 154)(29 155)(30 156)(31 144)(32 145)(33 146)(34 147)(35 148)(36 149)(37 150)(38 151)(39 152)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 118)(52 119)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 92)(65 93)(66 166)(67 167)(68 168)(69 169)(70 157)(71 158)(72 159)(73 160)(74 161)(75 162)(76 163)(77 164)(78 165)(79 172)(80 173)(81 174)(82 175)(83 176)(84 177)(85 178)(86 179)(87 180)(88 181)(89 182)(90 170)(91 171)(131 187)(132 188)(133 189)(134 190)(135 191)(136 192)(137 193)(138 194)(139 195)(140 183)(141 184)(142 185)(143 186)

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

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

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

260 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 13A ··· 13L 26A ··· 26AJ 26AK ··· 26BH 52A ··· 52AV 52AW ··· 52FL order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 1 ··· 1 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 C2 C4 C13 C26 C26 C26 C26 C52 C4○D4 C13×C4○D4 kernel C13×C42⋊C2 C4×C52 C13×C22⋊C4 C13×C4⋊C4 C22×C52 C2×C52 C42⋊C2 C42 C22⋊C4 C4⋊C4 C22×C4 C2×C4 C26 C2 # reps 1 2 2 2 1 8 12 24 24 24 12 96 4 48

Matrix representation of C13×C42⋊C2 in GL3(𝔽53) generated by

 1 0 0 0 13 0 0 0 13
,
 23 0 0 0 52 51 0 1 1
,
 52 0 0 0 30 0 0 0 30
,
 1 0 0 0 52 0 0 1 1
G:=sub<GL(3,GF(53))| [1,0,0,0,13,0,0,0,13],[23,0,0,0,52,1,0,51,1],[52,0,0,0,30,0,0,0,30],[1,0,0,0,52,1,0,0,1] >;

C13×C42⋊C2 in GAP, Magma, Sage, TeX

C_{13}\times C_4^2\rtimes C_2
% in TeX

G:=Group("C13xC4^2:C2");
// GroupNames label

G:=SmallGroup(416,178);
// by ID

G=gap.SmallGroup(416,178);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1248,1273,482]);
// Polycyclic

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

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