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G = C52.C8order 416 = 25·13

1st non-split extension by C52 of C8 acting via C8/C2=C4

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C52.1C8, C132M5(2), C4.(C13⋊C8), C13⋊C162C2, (C2×C52).7C4, (C2×C26).2C8, C26.6(C2×C8), C22.(C13⋊C8), C132C8.6C4, C52.18(C2×C4), C132C8.17C22, C2.3(C2×C13⋊C8), C4.19(C2×C13⋊C4), (C2×C4).5(C13⋊C4), (C2×C132C8).11C2, SmallGroup(416,73)

Series: Derived Chief Lower central Upper central

C1C26 — C52.C8
C1C13C26C52C132C8C13⋊C16 — C52.C8
C13C26 — C52.C8
C1C4C2×C4

Generators and relations for C52.C8
 G = < a,b | a52=1, b8=a26, bab-1=a31 >

2C2
2C26
13C8
13C8
13C16
13C2×C8
13C16
13M5(2)

Smallest permutation representation of C52.C8
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)
(1 183 75 144 14 170 88 131 27 157 101 118 40 196 62 105)(2 178 100 123 15 165 61 110 28 204 74 149 41 191 87 136)(3 173 73 154 16 160 86 141 29 199 99 128 42 186 60 115)(4 168 98 133 17 207 59 120 30 194 72 107 43 181 85 146)(5 163 71 112 18 202 84 151 31 189 97 138 44 176 58 125)(6 158 96 143 19 197 57 130 32 184 70 117 45 171 83 156)(7 205 69 122 20 192 82 109 33 179 95 148 46 166 56 135)(8 200 94 153 21 187 55 140 34 174 68 127 47 161 81 114)(9 195 67 132 22 182 80 119 35 169 93 106 48 208 54 145)(10 190 92 111 23 177 53 150 36 164 66 137 49 203 79 124)(11 185 65 142 24 172 78 129 37 159 91 116 50 198 104 155)(12 180 90 121 25 167 103 108 38 206 64 147 51 193 77 134)(13 175 63 152 26 162 76 139 39 201 89 126 52 188 102 113)

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

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

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

44 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F13A13B13C16A···16H26A···26I52A···52L
order12244488888813131316···1626···2652···52
size11211213131313262644426···264···44···4

44 irreducible representations

dim1111111244444
type++++-+-
imageC1C2C2C4C4C8C8M5(2)C13⋊C4C13⋊C8C2×C13⋊C4C13⋊C8C52.C8
kernelC52.C8C13⋊C16C2×C132C8C132C8C2×C52C52C2×C26C13C2×C4C4C4C22C1
# reps12122444333312

Matrix representation of C52.C8 in GL6(𝔽1249)

58510140000
06640000
00756101204564
00640101248
0016405661247
0026405661247
,
8612540000
8323880000
00909798346907
00583562447865
00853107525645
0024011411127502

G:=sub<GL(6,GF(1249))| [585,0,0,0,0,0,1014,664,0,0,0,0,0,0,75,640,1,2,0,0,610,1,640,640,0,0,1204,0,566,566,0,0,564,1248,1247,1247],[861,832,0,0,0,0,254,388,0,0,0,0,0,0,909,583,853,240,0,0,798,562,107,1141,0,0,346,447,525,1127,0,0,907,865,645,502] >;

C52.C8 in GAP, Magma, Sage, TeX

C_{52}.C_8
% in TeX

G:=Group("C52.C8");
// GroupNames label

G:=SmallGroup(416,73);
// by ID

G=gap.SmallGroup(416,73);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,50,69,9221,3473]);
// Polycyclic

G:=Group<a,b|a^52=1,b^8=a^26,b*a*b^-1=a^31>;
// generators/relations

Export

Subgroup lattice of C52.C8 in TeX

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