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G = C8.6D26order 416 = 25·13

3rd non-split extension by C8 of D26 acting via D26/D13=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.6D26, C52.5D4, C133SD32, Q161D13, C26.10D8, D104.2C2, C104.4C22, C132C163C2, (C13×Q16)⋊1C2, C2.6(D4⋊D13), C4.3(C13⋊D4), SmallGroup(416,35)

Series: Derived Chief Lower central Upper central

C1C104 — C8.6D26
C1C13C26C52C104D104 — C8.6D26
C13C26C52C104 — C8.6D26
C1C2C4C8Q16

Generators and relations for C8.6D26
 G = < a,b,c | a8=1, b26=a4, c2=a3, bab-1=a-1, ac=ca, cbc-1=a-1b25 >

104C2
4C4
52C22
8D13
2Q8
26D4
4D26
4C52
13C16
13D8
2D52
2Q8×C13
13SD32

Smallest permutation representation of C8.6D26
On 208 points
Generators in S208
(1 189 132 80 27 163 106 54)(2 55 107 164 28 81 133 190)(3 191 134 82 29 165 108 56)(4 57 109 166 30 83 135 192)(5 193 136 84 31 167 110 58)(6 59 111 168 32 85 137 194)(7 195 138 86 33 169 112 60)(8 61 113 170 34 87 139 196)(9 197 140 88 35 171 114 62)(10 63 115 172 36 89 141 198)(11 199 142 90 37 173 116 64)(12 65 117 174 38 91 143 200)(13 201 144 92 39 175 118 66)(14 67 119 176 40 93 145 202)(15 203 146 94 41 177 120 68)(16 69 121 178 42 95 147 204)(17 205 148 96 43 179 122 70)(18 71 123 180 44 97 149 206)(19 207 150 98 45 181 124 72)(20 73 125 182 46 99 151 208)(21 157 152 100 47 183 126 74)(22 75 127 184 48 101 153 158)(23 159 154 102 49 185 128 76)(24 77 129 186 50 103 155 160)(25 161 156 104 51 187 130 78)(26 79 131 188 52 53 105 162)
(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 26 80 188 106 105 189 79 27 52 54 162 132 131 163 53)(2 104 164 130 133 161 55 51 28 78 190 156 107 187 81 25)(3 24 82 186 108 155 191 77 29 50 56 160 134 129 165 103)(4 102 166 128 135 159 57 49 30 76 192 154 109 185 83 23)(5 22 84 184 110 153 193 75 31 48 58 158 136 127 167 101)(6 100 168 126 137 157 59 47 32 74 194 152 111 183 85 21)(7 20 86 182 112 151 195 73 33 46 60 208 138 125 169 99)(8 98 170 124 139 207 61 45 34 72 196 150 113 181 87 19)(9 18 88 180 114 149 197 71 35 44 62 206 140 123 171 97)(10 96 172 122 141 205 63 43 36 70 198 148 115 179 89 17)(11 16 90 178 116 147 199 69 37 42 64 204 142 121 173 95)(12 94 174 120 143 203 65 41 38 68 200 146 117 177 91 15)(13 14 92 176 118 145 201 67 39 40 66 202 144 119 175 93)

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

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

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

53 conjugacy classes

class 1 2A2B4A4B8A8B13A···13F16A16B16C16D26A···26F52A···52F52G···52R104A···104L
order122448813···131616161626···2652···5252···52104···104
size1110428222···2262626262···24···48···84···4

53 irreducible representations

dim111122222244
type++++++++++
imageC1C2C2C2D4D8D13SD32D26C13⋊D4D4⋊D13C8.6D26
kernelC8.6D26C132C16D104C13×Q16C52C26Q16C13C8C4C2C1
# reps11111264612612

Matrix representation of C8.6D26 in GL4(𝔽1249) generated by

1248000
0124800
0011651152
00850134
,
38421700
1811100
00785165
00686464
,
15753700
817109200
0011611084
00184785
G:=sub<GL(4,GF(1249))| [1248,0,0,0,0,1248,0,0,0,0,1165,850,0,0,1152,134],[384,18,0,0,217,111,0,0,0,0,785,686,0,0,165,464],[157,817,0,0,537,1092,0,0,0,0,1161,184,0,0,1084,785] >;

C8.6D26 in GAP, Magma, Sage, TeX

C_8._6D_{26}
% in TeX

G:=Group("C8.6D26");
// GroupNames label

G:=SmallGroup(416,35);
// by ID

G=gap.SmallGroup(416,35);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,73,103,218,116,122,579,297,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^26=a^4,c^2=a^3,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^25>;
// generators/relations

Export

Subgroup lattice of C8.6D26 in TeX

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