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G = C208⋊C2order 416 = 25·13

4th semidirect product of C208 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C2084C2, C163D13, D26.1C8, C8.20D26, C133M5(2), Dic13.1C8, C104.20C22, C132C164C2, C2.3(C8×D13), C132C8.2C4, C52.43(C2×C4), C26.11(C2×C8), (C4×D13).2C4, (C8×D13).2C2, C4.17(C4×D13), SmallGroup(416,5)

Series: Derived Chief Lower central Upper central

C1C26 — C208⋊C2
C1C13C26C52C104C8×D13 — C208⋊C2
C13C26 — C208⋊C2
C1C8C16

Generators and relations for C208⋊C2
 G = < a,b | a208=b2=1, bab=a25 >

26C2
13C4
13C22
2D13
13C2×C4
13C8
13C2×C8
13C16
13M5(2)

Smallest permutation representation of C208⋊C2
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)
(2 26)(3 51)(4 76)(5 101)(6 126)(7 151)(8 176)(9 201)(10 18)(11 43)(12 68)(13 93)(14 118)(15 143)(16 168)(17 193)(19 35)(20 60)(21 85)(22 110)(23 135)(24 160)(25 185)(28 52)(29 77)(30 102)(31 127)(32 152)(33 177)(34 202)(36 44)(37 69)(38 94)(39 119)(40 144)(41 169)(42 194)(45 61)(46 86)(47 111)(48 136)(49 161)(50 186)(54 78)(55 103)(56 128)(57 153)(58 178)(59 203)(62 70)(63 95)(64 120)(65 145)(66 170)(67 195)(71 87)(72 112)(73 137)(74 162)(75 187)(80 104)(81 129)(82 154)(83 179)(84 204)(88 96)(89 121)(90 146)(91 171)(92 196)(97 113)(98 138)(99 163)(100 188)(106 130)(107 155)(108 180)(109 205)(114 122)(115 147)(116 172)(117 197)(123 139)(124 164)(125 189)(132 156)(133 181)(134 206)(140 148)(141 173)(142 198)(149 165)(150 190)(158 182)(159 207)(166 174)(167 199)(175 191)(184 208)(192 200)

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

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

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

116 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F13A···13F16A16B16C16D16E16F16G16H26A···26F52A···52L104A···104X208A···208AV
order12244488888813···13161616161616161626···2652···52104···104208···208
size11261126111126262···22222262626262···22···22···22···2

116 irreducible representations

dim11111111222222
type++++++
imageC1C2C2C2C4C4C8C8D13M5(2)D26C4×D13C8×D13C208⋊C2
kernelC208⋊C2C132C16C208C8×D13C132C8C4×D13Dic13D26C16C13C8C4C2C1
# reps11112244646122448

Matrix representation of C208⋊C2 in GL2(𝔽1249) generated by

72954
808684
,
987393
426262
G:=sub<GL(2,GF(1249))| [72,808,954,684],[987,426,393,262] >;

C208⋊C2 in GAP, Magma, Sage, TeX

C_{208}\rtimes C_2
% in TeX

G:=Group("C208:C2");
// GroupNames label

G:=SmallGroup(416,5);
// by ID

G=gap.SmallGroup(416,5);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,31,50,69,13829]);
// Polycyclic

G:=Group<a,b|a^208=b^2=1,b*a*b=a^25>;
// generators/relations

Export

Subgroup lattice of C208⋊C2 in TeX

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