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## G = C13×C4⋊1D4order 416 = 25·13

### Direct product of C13 and C4⋊1D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C13×C4⋊1D4
 Chief series C1 — C2 — C22 — C2×C26 — C22×C26 — D4×C26 — C13×C4⋊1D4
 Lower central C1 — C22 — C13×C4⋊1D4
 Upper central C1 — C2×C26 — C13×C4⋊1D4

Generators and relations for C13×C41D4
G = < a,b,c,d | a13=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 180 in 108 conjugacy classes, 52 normal (8 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C22, C22 [×12], C2×C4 [×3], D4 [×12], C23 [×4], C13, C42, C2×D4 [×6], C26 [×3], C26 [×4], C41D4, C52 [×6], C2×C26, C2×C26 [×12], C2×C52 [×3], D4×C13 [×12], C22×C26 [×4], C4×C52, D4×C26 [×6], C13×C41D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C13, C2×D4 [×3], C26 [×7], C41D4, C2×C26 [×7], D4×C13 [×6], C22×C26, D4×C26 [×3], C13×C41D4

Smallest permutation representation of C13×C41D4
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 21 53 42)(2 22 54 43)(3 23 55 44)(4 24 56 45)(5 25 57 46)(6 26 58 47)(7 14 59 48)(8 15 60 49)(9 16 61 50)(10 17 62 51)(11 18 63 52)(12 19 64 40)(13 20 65 41)(27 99 192 70)(28 100 193 71)(29 101 194 72)(30 102 195 73)(31 103 183 74)(32 104 184 75)(33 92 185 76)(34 93 186 77)(35 94 187 78)(36 95 188 66)(37 96 189 67)(38 97 190 68)(39 98 191 69)(79 157 135 122)(80 158 136 123)(81 159 137 124)(82 160 138 125)(83 161 139 126)(84 162 140 127)(85 163 141 128)(86 164 142 129)(87 165 143 130)(88 166 131 118)(89 167 132 119)(90 168 133 120)(91 169 134 121)(105 155 179 206)(106 156 180 207)(107 144 181 208)(108 145 182 196)(109 146 170 197)(110 147 171 198)(111 148 172 199)(112 149 173 200)(113 150 174 201)(114 151 175 202)(115 152 176 203)(116 153 177 204)(117 154 178 205)
(1 86 189 180)(2 87 190 181)(3 88 191 182)(4 89 192 170)(5 90 193 171)(6 91 194 172)(7 79 195 173)(8 80 183 174)(9 81 184 175)(10 82 185 176)(11 83 186 177)(12 84 187 178)(13 85 188 179)(14 157 73 200)(15 158 74 201)(16 159 75 202)(17 160 76 203)(18 161 77 204)(19 162 78 205)(20 163 66 206)(21 164 67 207)(22 165 68 208)(23 166 69 196)(24 167 70 197)(25 168 71 198)(26 169 72 199)(27 109 56 132)(28 110 57 133)(29 111 58 134)(30 112 59 135)(31 113 60 136)(32 114 61 137)(33 115 62 138)(34 116 63 139)(35 117 64 140)(36 105 65 141)(37 106 53 142)(38 107 54 143)(39 108 55 131)(40 127 94 154)(41 128 95 155)(42 129 96 156)(43 130 97 144)(44 118 98 145)(45 119 99 146)(46 120 100 147)(47 121 101 148)(48 122 102 149)(49 123 103 150)(50 124 104 151)(51 125 92 152)(52 126 93 153)
(1 129)(2 130)(3 118)(4 119)(5 120)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 141)(21 142)(22 143)(23 131)(24 132)(25 133)(26 134)(27 197)(28 198)(29 199)(30 200)(31 201)(32 202)(33 203)(34 204)(35 205)(36 206)(37 207)(38 208)(39 196)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 79)(49 80)(50 81)(51 82)(52 83)(53 164)(54 165)(55 166)(56 167)(57 168)(58 169)(59 157)(60 158)(61 159)(62 160)(63 161)(64 162)(65 163)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(92 176)(93 177)(94 178)(95 179)(96 180)(97 181)(98 182)(99 170)(100 171)(101 172)(102 173)(103 174)(104 175)(144 190)(145 191)(146 192)(147 193)(148 194)(149 195)(150 183)(151 184)(152 185)(153 186)(154 187)(155 188)(156 189)

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

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

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

182 conjugacy classes

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

182 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C13 C26 C26 D4 D4×C13 kernel C13×C4⋊1D4 C4×C52 D4×C26 C4⋊1D4 C42 C2×D4 C52 C4 # reps 1 1 6 12 12 72 6 72

Matrix representation of C13×C41D4 in GL4(𝔽53) generated by

 16 0 0 0 0 16 0 0 0 0 13 0 0 0 0 13
,
 1 0 0 0 0 1 0 0 0 0 16 51 0 0 49 37
,
 0 1 0 0 52 0 0 0 0 0 1 0 0 0 0 1
,
 0 52 0 0 52 0 0 0 0 0 16 51 0 0 48 37
G:=sub<GL(4,GF(53))| [16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,16,49,0,0,51,37],[0,52,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,52,0,0,52,0,0,0,0,0,16,48,0,0,51,37] >;

C13×C41D4 in GAP, Magma, Sage, TeX

C_{13}\times C_4\rtimes_1D_4
% in TeX

G:=Group("C13xC4:1D4");
// GroupNames label

G:=SmallGroup(416,188);
// by ID

G=gap.SmallGroup(416,188);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,631,3818,950]);
// 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^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽