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G = C26.D4order 208 = 24·13

1st non-split extension by C26 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C26.5D4, C26.1Q8, Dic131C4, C2.1Dic26, C22.4D26, C132(C4⋊C4), (C2×C52).1C2, C2.4(C4×D13), (C2×C4).1D13, C26.11(C2×C4), C2.1(C13⋊D4), (C2×C26).4C22, (C2×Dic13).1C2, SmallGroup(208,12)

Series: Derived Chief Lower central Upper central

C1C26 — C26.D4
C1C13C26C2×C26C2×Dic13 — C26.D4
C13C26 — C26.D4
C1C22C2×C4

Generators and relations for C26.D4
 G = < a,b,c | a26=b4=1, c2=a13, bab-1=cac-1=a-1, cbc-1=b-1 >

2C4
13C4
13C4
26C4
13C2×C4
13C2×C4
2C52
2Dic13
13C4⋊C4

Smallest permutation representation of C26.D4
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 57 188 170)(2 56 189 169)(3 55 190 168)(4 54 191 167)(5 53 192 166)(6 78 193 165)(7 77 194 164)(8 76 195 163)(9 75 196 162)(10 74 197 161)(11 73 198 160)(12 72 199 159)(13 71 200 158)(14 70 201 157)(15 69 202 182)(16 68 203 181)(17 67 204 180)(18 66 205 179)(19 65 206 178)(20 64 207 177)(21 63 208 176)(22 62 183 175)(23 61 184 174)(24 60 185 173)(25 59 186 172)(26 58 187 171)(27 89 148 126)(28 88 149 125)(29 87 150 124)(30 86 151 123)(31 85 152 122)(32 84 153 121)(33 83 154 120)(34 82 155 119)(35 81 156 118)(36 80 131 117)(37 79 132 116)(38 104 133 115)(39 103 134 114)(40 102 135 113)(41 101 136 112)(42 100 137 111)(43 99 138 110)(44 98 139 109)(45 97 140 108)(46 96 141 107)(47 95 142 106)(48 94 143 105)(49 93 144 130)(50 92 145 129)(51 91 146 128)(52 90 147 127)
(1 82 14 95)(2 81 15 94)(3 80 16 93)(4 79 17 92)(5 104 18 91)(6 103 19 90)(7 102 20 89)(8 101 21 88)(9 100 22 87)(10 99 23 86)(11 98 24 85)(12 97 25 84)(13 96 26 83)(27 77 40 64)(28 76 41 63)(29 75 42 62)(30 74 43 61)(31 73 44 60)(32 72 45 59)(33 71 46 58)(34 70 47 57)(35 69 48 56)(36 68 49 55)(37 67 50 54)(38 66 51 53)(39 65 52 78)(105 189 118 202)(106 188 119 201)(107 187 120 200)(108 186 121 199)(109 185 122 198)(110 184 123 197)(111 183 124 196)(112 208 125 195)(113 207 126 194)(114 206 127 193)(115 205 128 192)(116 204 129 191)(117 203 130 190)(131 181 144 168)(132 180 145 167)(133 179 146 166)(134 178 147 165)(135 177 148 164)(136 176 149 163)(137 175 150 162)(138 174 151 161)(139 173 152 160)(140 172 153 159)(141 171 154 158)(142 170 155 157)(143 169 156 182)

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

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

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

C26.D4 is a maximal subgroup of
C4×Dic26  C52.6Q8  C42⋊D13  C422D13  C23.11D26  C22⋊Dic26  C23.D26  Dic134D4  D26.12D4  D26⋊D4  Dic133Q8  C52⋊Q8  Dic13.Q8  C4.Dic26  C4⋊C4×D13  D26.13D4  D26⋊Q8  C4⋊C4⋊D13  C52.48D4  C4×C13⋊D4  C23.23D26  C23.18D26  Dic13⋊D4  Dic13⋊Q8  D263Q8
C26.D4 is a maximal quotient of
C26.D8  C52.Q8  C52.8Q8  C52.53D4  C26.10C42

58 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F13A···13F26A···26R52A···52X
order122244444413···1326···2652···52
size111122262626262···22···22···2

58 irreducible representations

dim11112222222
type++++-++-
imageC1C2C2C4D4Q8D13D26Dic26C4×D13C13⋊D4
kernelC26.D4C2×Dic13C2×C52Dic13C26C26C2×C4C22C2C2C2
# reps12141166121212

Matrix representation of C26.D4 in GL3(𝔽53) generated by

5200
04747
0650
,
100
0711
03446
,
3000
01337
0440
G:=sub<GL(3,GF(53))| [52,0,0,0,47,6,0,47,50],[1,0,0,0,7,34,0,11,46],[30,0,0,0,13,4,0,37,40] >;

C26.D4 in GAP, Magma, Sage, TeX

C_{26}.D_4
% in TeX

G:=Group("C26.D4");
// GroupNames label

G:=SmallGroup(208,12);
// by ID

G=gap.SmallGroup(208,12);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,40,101,26,4804]);
// Polycyclic

G:=Group<a,b,c|a^26=b^4=1,c^2=a^13,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C26.D4 in TeX

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