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G = C104.6C4order 416 = 25·13

1st non-split extension by C104 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C104.6C4, C52.34D4, C4.18D52, C8.1Dic13, C22.2Dic26, (C2×C8).5D13, (C2×C26).3Q8, (C2×C104).7C2, C52.56(C2×C4), (C2×C4).70D26, C26.14(C4⋊C4), C133(C8.C4), C4.8(C2×Dic13), C2.5(C523C4), C52.4C4.1C2, (C2×C52).97C22, SmallGroup(416,26)

Series: Derived Chief Lower central Upper central

C1C52 — C104.6C4
C1C13C26C52C2×C52C52.4C4 — C104.6C4
C13C26C52 — C104.6C4
C1C4C2×C4C2×C8

Generators and relations for C104.6C4
 G = < a,b | a104=1, b4=a52, bab-1=a51 >

2C2
2C26
26C8
26C8
13M4(2)
13M4(2)
2C132C8
2C132C8
13C8.C4

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

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

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

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

110 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F8G8H13A···13F26A···26R52A···52X104A···104AV
order1224448888888813···1326···2652···52104···104
size1121122222525252522···22···22···22···2

110 irreducible representations

dim1111222222222
type++++-+-++-
imageC1C2C2C4D4Q8D13C8.C4Dic13D26D52Dic26C104.6C4
kernelC104.6C4C52.4C4C2×C104C104C52C2×C26C2×C8C13C8C2×C4C4C22C1
# reps12141164126121248

Matrix representation of C104.6C4 in GL2(𝔽313) generated by

400
0133
,
01
250
G:=sub<GL(2,GF(313))| [40,0,0,133],[0,25,1,0] >;

C104.6C4 in GAP, Magma, Sage, TeX

C_{104}._6C_4
% in TeX

G:=Group("C104.6C4");
// GroupNames label

G:=SmallGroup(416,26);
// by ID

G=gap.SmallGroup(416,26);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,55,86,579,69,13829]);
// Polycyclic

G:=Group<a,b|a^104=1,b^4=a^52,b*a*b^-1=a^51>;
// generators/relations

Export

Subgroup lattice of C104.6C4 in TeX

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