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G = C52.4C8order 416 = 25·13

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

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C52.4C8, C104.9C4, C8.22D26, C134M5(2), C8.2Dic13, C104.22C22, C4.(C132C8), (C2×C26).5C8, (C2×C8).7D13, C132C165C2, C26.18(C2×C8), (C2×C52).19C4, C52.60(C2×C4), (C2×C104).10C2, C22.(C132C8), (C2×C4).5Dic13, C4.11(C2×Dic13), C2.4(C2×C132C8), SmallGroup(416,19)

Series: Derived Chief Lower central Upper central

C1C26 — C52.4C8
C1C13C26C52C104C132C16 — C52.4C8
C13C26 — C52.4C8
C1C8C2×C8

Generators and relations for C52.4C8
 G = < a,b | a52=1, b8=a26, bab-1=a-1 >

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

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

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

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

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

116 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F13A···13F16A···16H26A···26R52A···52X104A···104AV
order12244488888813···1316···1626···2652···52104···104
size1121121111222···226···262···22···22···2

116 irreducible representations

dim111111122222222
type++++-+-
imageC1C2C2C4C4C8C8D13M5(2)Dic13D26Dic13C132C8C132C8C52.4C8
kernelC52.4C8C132C16C2×C104C104C2×C52C52C2×C26C2×C8C13C8C8C2×C4C4C22C1
# reps121224464666121248

Matrix representation of C52.4C8 in GL2(𝔽1249) generated by

1560
3161241
,
509305
300740
G:=sub<GL(2,GF(1249))| [156,316,0,1241],[509,300,305,740] >;

C52.4C8 in GAP, Magma, Sage, TeX

C_{52}._4C_8
% in TeX

G:=Group("C52.4C8");
// GroupNames label

G:=SmallGroup(416,19);
// by ID

G=gap.SmallGroup(416,19);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,50,69,13829]);
// Polycyclic

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

Export

Subgroup lattice of C52.4C8 in TeX

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