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G = C13×C4⋊C4order 208 = 24·13

Direct product of C13 and C4⋊C4

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

Aliases: C13×C4⋊C4, C4⋊C52, C525C4, C26.3Q8, C26.13D4, C2.(Q8×C13), C2.2(C2×C52), (C2×C52).2C2, (C2×C4).1C26, C2.2(D4×C13), C26.18(C2×C4), C22.3(C2×C26), (C2×C26).14C22, SmallGroup(208,22)

Series: Derived Chief Lower central Upper central

C1C2 — C13×C4⋊C4
C1C2C22C2×C26C2×C52 — C13×C4⋊C4
C1C2 — C13×C4⋊C4
C1C2×C26 — C13×C4⋊C4

Generators and relations for C13×C4⋊C4
 G = < a,b,c | a13=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C4
2C52
2C52

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

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

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

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

C13×C4⋊C4 is a maximal subgroup of
C26.D8  C52.Q8  D526C4  C26.Q16  Dic133Q8  C52⋊Q8  Dic13.Q8  C4.Dic26  C4⋊C47D13  D528C4  D26.13D4  C42D52  D26⋊Q8  D262Q8  C4⋊C4⋊D13  D4×C52  Q8×C52

130 conjugacy classes

class 1 2A2B2C4A···4F13A···13L26A···26AJ52A···52BT
order12224···413···1326···2652···52
size11112···21···11···12···2

130 irreducible representations

dim1111112222
type+++-
imageC1C2C4C13C26C52D4Q8D4×C13Q8×C13
kernelC13×C4⋊C4C2×C52C52C4⋊C4C2×C4C4C26C26C2C2
# reps134123648111212

Matrix representation of C13×C4⋊C4 in GL3(𝔽53) generated by

100
0460
0046
,
5200
001
0520
,
2300
03110
01022
G:=sub<GL(3,GF(53))| [1,0,0,0,46,0,0,0,46],[52,0,0,0,0,52,0,1,0],[23,0,0,0,31,10,0,10,22] >;

C13×C4⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C13xC4:C4");
// GroupNames label

G:=SmallGroup(208,22);
// by ID

G=gap.SmallGroup(208,22);
# by ID

G:=PCGroup([5,-2,-2,-13,-2,-2,520,541,266]);
// Polycyclic

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

Export

Subgroup lattice of C13×C4⋊C4 in TeX

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