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G = C2×C132C8order 208 = 24·13

Direct product of C2 and C132C8

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2×C132C8, C262C8, C52.6C4, C4.14D26, C4.3Dic13, C52.14C22, C22.2Dic13, C134(C2×C8), (C2×C52).6C2, (C2×C26).4C4, (C2×C4).5D13, C26.13(C2×C4), C2.1(C2×Dic13), SmallGroup(208,9)

Series: Derived Chief Lower central Upper central

C1C13 — C2×C132C8
C1C13C26C52C132C8 — C2×C132C8
C13 — C2×C132C8
C1C2×C4

Generators and relations for C2×C132C8
 G = < a,b,c | a2=b13=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

13C8
13C8
13C2×C8

Smallest permutation representation of C2×C132C8
Regular action on 208 points
Generators in S208
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)(105 161)(106 162)(107 163)(108 164)(109 165)(110 166)(111 167)(112 168)(113 169)(114 157)(115 158)(116 159)(117 160)(118 172)(119 173)(120 174)(121 175)(122 176)(123 177)(124 178)(125 179)(126 180)(127 181)(128 182)(129 170)(130 171)(131 193)(132 194)(133 195)(134 183)(135 184)(136 185)(137 186)(138 187)(139 188)(140 189)(141 190)(142 191)(143 192)(144 208)(145 196)(146 197)(147 198)(148 199)(149 200)(150 201)(151 202)(152 203)(153 204)(154 205)(155 206)(156 207)
(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)
(1 201 40 181 14 189 27 165)(2 200 41 180 15 188 28 164)(3 199 42 179 16 187 29 163)(4 198 43 178 17 186 30 162)(5 197 44 177 18 185 31 161)(6 196 45 176 19 184 32 160)(7 208 46 175 20 183 33 159)(8 207 47 174 21 195 34 158)(9 206 48 173 22 194 35 157)(10 205 49 172 23 193 36 169)(11 204 50 171 24 192 37 168)(12 203 51 170 25 191 38 167)(13 202 52 182 26 190 39 166)(53 150 92 127 66 140 79 109)(54 149 93 126 67 139 80 108)(55 148 94 125 68 138 81 107)(56 147 95 124 69 137 82 106)(57 146 96 123 70 136 83 105)(58 145 97 122 71 135 84 117)(59 144 98 121 72 134 85 116)(60 156 99 120 73 133 86 115)(61 155 100 119 74 132 87 114)(62 154 101 118 75 131 88 113)(63 153 102 130 76 143 89 112)(64 152 103 129 77 142 90 111)(65 151 104 128 78 141 91 110)

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

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

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

C2×C132C8 is a maximal subgroup of
C26.7C42  C523C8  C26.D8  C52.Q8  D526C4  C26.Q16  C8×Dic13  C52.8Q8  C1048C4  D261C8  C52.53D4  C52.55D4  D4⋊Dic13  Q8⋊Dic13  C52.C8  C2×C8×D13  D52.2C4  D4.Dic13  C52.C23
C2×C132C8 is a maximal quotient of
C523C8  C52.4C8  C52.55D4

64 conjugacy classes

class 1 2A2B2C4A4B4C4D8A···8H13A···13F26A···26R52A···52X
order122244448···813···1326···2652···52
size1111111113···132···22···22···2

64 irreducible representations

dim11111122222
type++++-+-
imageC1C2C2C4C4C8D13Dic13D26Dic13C132C8
kernelC2×C132C8C132C8C2×C52C52C2×C26C26C2×C4C4C4C22C2
# reps121228666624

Matrix representation of C2×C132C8 in GL3(𝔽313) generated by

31200
03120
00312
,
100
078312
079312
,
28800
0221
0252311
G:=sub<GL(3,GF(313))| [312,0,0,0,312,0,0,0,312],[1,0,0,0,78,79,0,312,312],[288,0,0,0,2,252,0,21,311] >;

C2×C132C8 in GAP, Magma, Sage, TeX

C_2\times C_{13}\rtimes_2C_8
% in TeX

G:=Group("C2xC13:2C8");
// GroupNames label

G:=SmallGroup(208,9);
// by ID

G=gap.SmallGroup(208,9);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,20,42,4804]);
// Polycyclic

G:=Group<a,b,c|a^2=b^13=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C2×C132C8 in TeX

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