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G = C104.C4order 416 = 25·13

2nd non-split extension by C104 of C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C104.2C4, D26.1Q8, Dic13.10D4, C8.2(C13⋊C4), C26.3(C4⋊C4), C132C8.3C4, C52.10(C2×C4), (C8×D13).6C2, C131(C8.C4), C2.6(C52⋊C4), C52.C4.1C2, (C4×D13).27C22, C4.10(C2×C13⋊C4), SmallGroup(416,70)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C104.C4
 G = < a,b | a104=1, b4=a52, bab-1=a83 >

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

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

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

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

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.C4
kernelC104.C4C8×D13C52.C4C132C8C104Dic13D26C13C8C4C2C1
# reps1122211433612

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

23225209137
10297126153
29435301148
248125210163
,
7213105223
27221728865
28215614810
5246307254
G:=sub<GL(4,GF(313))| [23,102,294,248,225,97,35,125,209,126,301,210,137,153,148,163],[7,272,282,52,213,217,156,46,105,288,148,307,223,65,10,254] >;

C104.C4 in GAP, Magma, Sage, TeX

C_{104}.C_4
% in TeX

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

G:=SmallGroup(416,70);
// by ID

G=gap.SmallGroup(416,70);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,55,86,579,69,9221,3473]);
// Polycyclic

G:=Group<a,b|a^104=1,b^4=a^52,b*a*b^-1=a^83>;
// generators/relations

Export

Subgroup lattice of C104.C4 in TeX

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