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G = C523C4order 208 = 24·13

1st semidirect product of C52 and C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C523C4, C4⋊Dic13, C26.4D4, C2.1D52, C26.2Q8, C2.2Dic26, C22.5D26, C133(C4⋊C4), (C2×C52).3C2, (C2×C4).3D13, C26.15(C2×C4), (C2×C26).5C22, C2.4(C2×Dic13), (C2×Dic13).2C2, SmallGroup(208,13)

Series: Derived Chief Lower central Upper central

C1C26 — C523C4
C1C13C26C2×C26C2×Dic13 — C523C4
C13C26 — C523C4
C1C22C2×C4

Generators and relations for C523C4
 G = < a,b | a52=b4=1, bab-1=a-1 >

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

Smallest permutation representation of C523C4
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)
(1 123 175 78)(2 122 176 77)(3 121 177 76)(4 120 178 75)(5 119 179 74)(6 118 180 73)(7 117 181 72)(8 116 182 71)(9 115 183 70)(10 114 184 69)(11 113 185 68)(12 112 186 67)(13 111 187 66)(14 110 188 65)(15 109 189 64)(16 108 190 63)(17 107 191 62)(18 106 192 61)(19 105 193 60)(20 156 194 59)(21 155 195 58)(22 154 196 57)(23 153 197 56)(24 152 198 55)(25 151 199 54)(26 150 200 53)(27 149 201 104)(28 148 202 103)(29 147 203 102)(30 146 204 101)(31 145 205 100)(32 144 206 99)(33 143 207 98)(34 142 208 97)(35 141 157 96)(36 140 158 95)(37 139 159 94)(38 138 160 93)(39 137 161 92)(40 136 162 91)(41 135 163 90)(42 134 164 89)(43 133 165 88)(44 132 166 87)(45 131 167 86)(46 130 168 85)(47 129 169 84)(48 128 170 83)(49 127 171 82)(50 126 172 81)(51 125 173 80)(52 124 174 79)

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

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

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

C523C4 is a maximal subgroup of
C26.D8  C52.Q8  C52.44D4  C1046C4  C1045C4  D525C4  D4⋊Dic13  Q8⋊Dic13  C4×Dic26  C522Q8  C52.6Q8  C4×D52  C22⋊Dic26  C23.D26  D26.12D4  C22.D52  C52⋊Q8  Dic13.Q8  C4.Dic26  C4⋊C4×D13  C4⋊C47D13  D262Q8  C4⋊C4⋊D13  C52.48D4  C23.21D26  C527D4  D4×Dic13  C522D4  Q8×Dic13  D263Q8
C523C4 is a maximal quotient of
C523C8  C1046C4  C1045C4  C104.6C4  C26.10C42

58 conjugacy classes

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

58 irreducible representations

dim11112222222
type++++-+-+-+
imageC1C2C2C4D4Q8D13Dic13D26Dic26D52
kernelC523C4C2×Dic13C2×C52C52C26C26C2×C4C4C22C2C2
# reps12141161261212

Matrix representation of C523C4 in GL3(𝔽53) generated by

100
05038
03131
,
2300
01035
03243
G:=sub<GL(3,GF(53))| [1,0,0,0,50,31,0,38,31],[23,0,0,0,10,32,0,35,43] >;

C523C4 in GAP, Magma, Sage, TeX

C_{52}\rtimes_3C_4
% in TeX

G:=Group("C52:3C4");
// GroupNames label

G:=SmallGroup(208,13);
// by ID

G=gap.SmallGroup(208,13);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,20,101,46,4804]);
// Polycyclic

G:=Group<a,b|a^52=b^4=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C523C4 in TeX

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