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## G = C13×C4⋊D4order 416 = 25·13

### Direct product of C13 and C4⋊D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C13×C4⋊D4
 Chief series C1 — C2 — C22 — C2×C26 — C22×C26 — D4×C26 — C13×C4⋊D4
 Lower central C1 — C22 — C13×C4⋊D4
 Upper central C1 — C2×C26 — C13×C4⋊D4

Generators and relations for C13×C4⋊D4
G = < a,b,c,d | a13=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 148 in 94 conjugacy classes, 48 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C2×C4 [×2], C2×C4 [×2], C2×C4 [×2], D4 [×6], C23, C23 [×2], C13, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], C26 [×3], C26 [×4], C4⋊D4, C52 [×2], C52 [×3], C2×C26, C2×C26 [×2], C2×C26 [×8], C2×C52 [×2], C2×C52 [×2], C2×C52 [×2], D4×C13 [×6], C22×C26, C22×C26 [×2], C13×C22⋊C4 [×2], C13×C4⋊C4, C22×C52, D4×C26, D4×C26 [×2], C13×C4⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C13, C2×D4 [×2], C4○D4, C26 [×7], C4⋊D4, C2×C26 [×7], D4×C13 [×4], C22×C26, D4×C26 [×2], C13×C4○D4, C13×C4⋊D4

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

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

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

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

182 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 13A ··· 13L 26A ··· 26AJ 26AK ··· 26BH 26BI ··· 26CF 52A ··· 52AV 52AW ··· 52BT order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 13 ··· 13 26 ··· 26 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 1 1 2 2 4 4 2 2 2 2 4 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

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 C4○D4 D4×C13 D4×C13 C13×C4○D4 kernel C13×C4⋊D4 C13×C22⋊C4 C13×C4⋊C4 C22×C52 D4×C26 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C52 C2×C26 C26 C4 C22 C2 # reps 1 2 1 1 3 12 24 12 12 36 2 2 2 24 24 24

Matrix representation of C13×C4⋊D4 in GL4(𝔽53) generated by

 28 0 0 0 0 28 0 0 0 0 42 0 0 0 0 42
,
 23 44 0 0 0 30 0 0 0 0 0 1 0 0 52 0
,
 24 47 0 0 52 29 0 0 0 0 10 22 0 0 22 43
,
 29 8 0 0 1 24 0 0 0 0 10 22 0 0 22 43
G:=sub<GL(4,GF(53))| [28,0,0,0,0,28,0,0,0,0,42,0,0,0,0,42],[23,0,0,0,44,30,0,0,0,0,0,52,0,0,1,0],[24,52,0,0,47,29,0,0,0,0,10,22,0,0,22,43],[29,1,0,0,8,24,0,0,0,0,10,22,0,0,22,43] >;

C13×C4⋊D4 in GAP, Magma, Sage, TeX

C_{13}\times C_4\rtimes D_4
% in TeX

G:=Group("C13xC4:D4");
// GroupNames label

G:=SmallGroup(416,182);
// by ID

G=gap.SmallGroup(416,182);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,631,3818]);
// 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,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽