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G = C13×D4⋊C4order 416 = 25·13

Direct product of C13 and D4⋊C4

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

Aliases: C13×D4⋊C4, D41C52, C26.13D8, C52.60D4, C26.9SD16, C4⋊C41C26, (C2×C8)⋊2C26, (C2×C104)⋊4C2, (D4×C13)⋊7C4, C4.1(C2×C52), C2.1(C13×D8), C52.49(C2×C4), (C2×D4).3C26, (D4×C26).9C2, C4.11(D4×C13), (C2×C26).46D4, C2.1(C13×SD16), C22.8(D4×C13), C26.35(C22⋊C4), (C2×C52).114C22, (C13×C4⋊C4)⋊10C2, (C2×C4).17(C2×C26), C2.6(C13×C22⋊C4), SmallGroup(416,52)

Series: Derived Chief Lower central Upper central

C1C4 — C13×D4⋊C4
C1C2C22C2×C4C2×C52C13×C4⋊C4 — C13×D4⋊C4
C1C2C4 — C13×D4⋊C4
C1C2×C26C2×C52 — C13×D4⋊C4

Generators and relations for C13×D4⋊C4
 G = < a,b,c,d | a13=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

4C2
4C2
2C22
2C22
4C22
4C22
4C4
4C26
4C26
2C2×C4
2C23
2C8
2D4
2C2×C26
2C2×C26
4C2×C26
4C52
4C2×C26
2C104
2C22×C26
2C2×C52
2D4×C13

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

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

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

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

182 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D8A8B8C8D13A···13L26A···26AJ26AK···26BH52A···52X52Y···52AV104A···104AV
order1222224444888813···1326···2626···2652···5252···52104···104
size111144224422221···11···14···42···24···42···2

182 irreducible representations

dim111111111122222222
type+++++++
imageC1C2C2C2C4C13C26C26C26C52D4D4D8SD16D4×C13D4×C13C13×D8C13×SD16
kernelC13×D4⋊C4C13×C4⋊C4C2×C104D4×C26D4×C13D4⋊C4C4⋊C4C2×C8C2×D4D4C52C2×C26C26C26C4C22C2C2
# reps111141212121248112212122424

Matrix representation of C13×D4⋊C4 in GL3(𝔽313) generated by

100
0440
0044
,
100
01311
01312
,
100
01311
00312
,
28800
00193
02530
G:=sub<GL(3,GF(313))| [1,0,0,0,44,0,0,0,44],[1,0,0,0,1,1,0,311,312],[1,0,0,0,1,0,0,311,312],[288,0,0,0,0,253,0,193,0] >;

C13×D4⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C13xD4:C4");
// GroupNames label

G:=SmallGroup(416,52);
// by ID

G=gap.SmallGroup(416,52);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,6243,3129,117]);
// Polycyclic

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

Export

Subgroup lattice of C13×D4⋊C4 in TeX

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