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G = C4×C52order 208 = 24·13

Abelian group of type [4,52]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C52, SmallGroup(208,20)

Series: Derived Chief Lower central Upper central

C1 — C4×C52
C1C2C22C2×C26C2×C52 — C4×C52
C1 — C4×C52
C1 — C4×C52

Generators and relations for C4×C52
 G = < a,b | a4=b52=1, ab=ba >


Smallest permutation representation of C4×C52
Regular action on 208 points
Generators in S208
(1 201 136 78)(2 202 137 79)(3 203 138 80)(4 204 139 81)(5 205 140 82)(6 206 141 83)(7 207 142 84)(8 208 143 85)(9 157 144 86)(10 158 145 87)(11 159 146 88)(12 160 147 89)(13 161 148 90)(14 162 149 91)(15 163 150 92)(16 164 151 93)(17 165 152 94)(18 166 153 95)(19 167 154 96)(20 168 155 97)(21 169 156 98)(22 170 105 99)(23 171 106 100)(24 172 107 101)(25 173 108 102)(26 174 109 103)(27 175 110 104)(28 176 111 53)(29 177 112 54)(30 178 113 55)(31 179 114 56)(32 180 115 57)(33 181 116 58)(34 182 117 59)(35 183 118 60)(36 184 119 61)(37 185 120 62)(38 186 121 63)(39 187 122 64)(40 188 123 65)(41 189 124 66)(42 190 125 67)(43 191 126 68)(44 192 127 69)(45 193 128 70)(46 194 129 71)(47 195 130 72)(48 196 131 73)(49 197 132 74)(50 198 133 75)(51 199 134 76)(52 200 135 77)
(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)

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

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

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

C4×C52 is a maximal subgroup of
C26.7C42  C523C8  D524C4  C522Q8  C52.6Q8  C42⋊D13  C4⋊D52  C4.D52  C422D13

208 conjugacy classes

class 1 2A2B2C4A···4L13A···13L26A···26AJ52A···52EN
order12224···413···1326···2652···52
size11111···11···11···11···1

208 irreducible representations

dim111111
type++
imageC1C2C4C13C26C52
kernelC4×C52C2×C52C52C42C2×C4C4
# reps13121236144

Matrix representation of C4×C52 in GL2(𝔽53) generated by

230
023
,
190
028
G:=sub<GL(2,GF(53))| [23,0,0,23],[19,0,0,28] >;

C4×C52 in GAP, Magma, Sage, TeX

C_4\times C_{52}
% in TeX

G:=Group("C4xC52");
// GroupNames label

G:=SmallGroup(208,20);
// by ID

G=gap.SmallGroup(208,20);
# by ID

G:=PCGroup([5,-2,-2,-13,-2,-2,260,526]);
// Polycyclic

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

Export

Subgroup lattice of C4×C52 in TeX

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