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## G = C13×C4.4D4order 416 = 25·13

### Direct product of C13 and C4.4D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C13×C4.4D4
 Chief series C1 — C2 — C22 — C2×C26 — C22×C26 — C13×C22⋊C4 — C13×C4.4D4
 Lower central C1 — C22 — C13×C4.4D4
 Upper central C1 — C2×C26 — C13×C4.4D4

Generators and relations for C13×C4.4D4
G = < a,b,c,d | a13=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 116 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C13, C42, C22⋊C4 [×4], C2×D4, C2×Q8, C26, C26 [×2], C26 [×2], C4.4D4, C52 [×2], C52 [×4], C2×C26, C2×C26 [×6], C2×C52, C2×C52 [×4], D4×C13 [×2], Q8×C13 [×2], C22×C26 [×2], C4×C52, C13×C22⋊C4 [×4], D4×C26, Q8×C26, C13×C4.4D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C13, C2×D4, C4○D4 [×2], C26 [×7], C4.4D4, C2×C26 [×7], D4×C13 [×2], C22×C26, D4×C26, C13×C4○D4 [×2], C13×C4.4D4

Smallest permutation representation of C13×C4.4D4
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)
(1 136 87 170)(2 137 88 171)(3 138 89 172)(4 139 90 173)(5 140 91 174)(6 141 79 175)(7 142 80 176)(8 143 81 177)(9 131 82 178)(10 132 83 179)(11 133 84 180)(12 134 85 181)(13 135 86 182)(14 153 59 115)(15 154 60 116)(16 155 61 117)(17 156 62 105)(18 144 63 106)(19 145 64 107)(20 146 65 108)(21 147 53 109)(22 148 54 110)(23 149 55 111)(24 150 56 112)(25 151 57 113)(26 152 58 114)(27 204 101 166)(28 205 102 167)(29 206 103 168)(30 207 104 169)(31 208 92 157)(32 196 93 158)(33 197 94 159)(34 198 95 160)(35 199 96 161)(36 200 97 162)(37 201 98 163)(38 202 99 164)(39 203 100 165)(40 194 129 66)(41 195 130 67)(42 183 118 68)(43 184 119 69)(44 185 120 70)(45 186 121 71)(46 187 122 72)(47 188 123 73)(48 189 124 74)(49 190 125 75)(50 191 126 76)(51 192 127 77)(52 193 128 78)
(1 47 55 27)(2 48 56 28)(3 49 57 29)(4 50 58 30)(5 51 59 31)(6 52 60 32)(7 40 61 33)(8 41 62 34)(9 42 63 35)(10 43 64 36)(11 44 65 37)(12 45 53 38)(13 46 54 39)(14 92 91 127)(15 93 79 128)(16 94 80 129)(17 95 81 130)(18 96 82 118)(19 97 83 119)(20 98 84 120)(21 99 85 121)(22 100 86 122)(23 101 87 123)(24 102 88 124)(25 103 89 125)(26 104 90 126)(66 155 159 176)(67 156 160 177)(68 144 161 178)(69 145 162 179)(70 146 163 180)(71 147 164 181)(72 148 165 182)(73 149 166 170)(74 150 167 171)(75 151 168 172)(76 152 169 173)(77 153 157 174)(78 154 158 175)(105 198 143 195)(106 199 131 183)(107 200 132 184)(108 201 133 185)(109 202 134 186)(110 203 135 187)(111 204 136 188)(112 205 137 189)(113 206 138 190)(114 207 139 191)(115 208 140 192)(116 196 141 193)(117 197 142 194)
(1 204 87 166)(2 205 88 167)(3 206 89 168)(4 207 90 169)(5 208 91 157)(6 196 79 158)(7 197 80 159)(8 198 81 160)(9 199 82 161)(10 200 83 162)(11 201 84 163)(12 202 85 164)(13 203 86 165)(14 77 59 192)(15 78 60 193)(16 66 61 194)(17 67 62 195)(18 68 63 183)(19 69 64 184)(20 70 65 185)(21 71 53 186)(22 72 54 187)(23 73 55 188)(24 74 56 189)(25 75 57 190)(26 76 58 191)(27 170 101 136)(28 171 102 137)(29 172 103 138)(30 173 104 139)(31 174 92 140)(32 175 93 141)(33 176 94 142)(34 177 95 143)(35 178 96 131)(36 179 97 132)(37 180 98 133)(38 181 99 134)(39 182 100 135)(40 155 129 117)(41 156 130 105)(42 144 118 106)(43 145 119 107)(44 146 120 108)(45 147 121 109)(46 148 122 110)(47 149 123 111)(48 150 124 112)(49 151 125 113)(50 152 126 114)(51 153 127 115)(52 154 128 116)

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

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

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

182 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4F 4G 4H 13A ··· 13L 26A ··· 26AJ 26AK ··· 26BH 52A ··· 52BT 52BU ··· 52CR order 1 2 2 2 2 2 4 ··· 4 4 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 1 1 4 4 2 ··· 2 4 4 1 ··· 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

182 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C13 C26 C26 C26 C26 D4 C4○D4 D4×C13 C13×C4○D4 kernel C13×C4.4D4 C4×C52 C13×C22⋊C4 D4×C26 Q8×C26 C4.4D4 C42 C22⋊C4 C2×D4 C2×Q8 C52 C26 C4 C2 # reps 1 1 4 1 1 12 12 48 12 12 2 4 24 48

Matrix representation of C13×C4.4D4 in GL4(𝔽53) generated by

 42 0 0 0 0 42 0 0 0 0 36 0 0 0 0 36
,
 10 44 0 0 23 43 0 0 0 0 1 0 0 0 0 1
,
 35 48 0 0 1 18 0 0 0 0 13 51 0 0 32 40
,
 23 33 0 0 0 30 0 0 0 0 13 51 0 0 31 40
G:=sub<GL(4,GF(53))| [42,0,0,0,0,42,0,0,0,0,36,0,0,0,0,36],[10,23,0,0,44,43,0,0,0,0,1,0,0,0,0,1],[35,1,0,0,48,18,0,0,0,0,13,32,0,0,51,40],[23,0,0,0,33,30,0,0,0,0,13,31,0,0,51,40] >;

C13×C4.4D4 in GAP, Magma, Sage, TeX

C_{13}\times C_4._4D_4
% in TeX

G:=Group("C13xC4.4D4");
// GroupNames label

G:=SmallGroup(416,185);
// by ID

G=gap.SmallGroup(416,185);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,1255,3818,482]);
// Polycyclic

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

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