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G = C104.1C4order 416 = 25·13

1st non-split extension by C104 of C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C104.1C4, D26.2Q8, Dic13.11D4, C8.1(C13⋊C4), C26.4(C4⋊C4), C132C8.4C4, C52.11(C2×C4), (C8×D13).4C2, C132(C8.C4), C2.7(C52⋊C4), C52.C4.2C2, (C4×D13).28C22, C4.11(C2×C13⋊C4), SmallGroup(416,71)

Series: Derived Chief Lower central Upper central

C1C52 — C104.1C4
C1C13C26Dic13C4×D13C52.C4 — C104.1C4
C13C26C52 — C104.1C4
C1C2C4C8

Generators and relations for C104.1C4
 G = < a,b | a104=1, b4=a52, bab-1=a31 >

26C2
13C22
13C4
2D13
13C8
13C2×C4
26C8
26C8
13C2×C8
13M4(2)
13M4(2)
2C13⋊C8
2C13⋊C8
13C8.C4

Smallest permutation representation of C104.1C4
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)
(1 169 27 143 53 117 79 195)(2 112 52 174 54 164 104 122)(3 159 77 205 55 107 25 153)(4 206 102 132 56 154 50 184)(5 149 23 163 57 201 75 111)(6 196 48 194 58 144 100 142)(7 139 73 121 59 191 21 173)(8 186 98 152 60 134 46 204)(9 129 19 183 61 181 71 131)(10 176 44 110 62 124 96 162)(11 119 69 141 63 171 17 193)(12 166 94 172 64 114 42 120)(13 109 15 203 65 161 67 151)(14 156 40 130 66 208 92 182)(16 146 90 192 68 198 38 140)(18 136 36 150 70 188 88 202)(20 126 86 108 72 178 34 160)(22 116 32 170 74 168 84 118)(24 106 82 128 76 158 30 180)(26 200 28 190 78 148 80 138)(29 133 103 179 81 185 51 127)(31 123 49 137 83 175 101 189)(33 113 99 199 85 165 47 147)(35 207 45 157 87 155 97 105)(37 197 95 115 89 145 43 167)(39 187 41 177 91 135 93 125)

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

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

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

38 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F8G8H13A13B13C26A26B26C52A···52F104A···104L
order1224448888888813131326262652···52104···104
size112621313222626525252524444444···44···4

38 irreducible representations

dim111112224444
type++++-++
imageC1C2C2C4C4D4Q8C8.C4C13⋊C4C2×C13⋊C4C52⋊C4C104.1C4
kernelC104.1C4C8×D13C52.C4C132C8C104Dic13D26C13C8C4C2C1
# reps1122211433612

Matrix representation of C104.1C4 in GL4(𝔽313) generated by

191122251307
4221140132
42888548
191626279
,
213120116221
3817772250
241136275109
197193100274
G:=sub<GL(4,GF(313))| [191,42,42,191,122,211,88,62,251,40,85,6,307,132,48,279],[213,38,241,197,120,177,136,193,116,72,275,100,221,250,109,274] >;

C104.1C4 in GAP, Magma, Sage, TeX

C_{104}._1C_4
% in TeX

G:=Group("C104.1C4");
// GroupNames label

G:=SmallGroup(416,71);
// by ID

G=gap.SmallGroup(416,71);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,151,86,579,69,9221,3473]);
// Polycyclic

G:=Group<a,b|a^104=1,b^4=a^52,b*a*b^-1=a^31>;
// generators/relations

Export

Subgroup lattice of C104.1C4 in TeX

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