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G = C13×C8.C4order 416 = 25·13

Direct product of C13 and C8.C4

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C13×C8.C4, C8.1C52, C52.68D4, C104.10C4, M4(2).2C26, C4.8(C2×C52), (C2×C8).5C26, (C2×C26).2Q8, C52.66(C2×C4), C22.(Q8×C13), C4.19(D4×C13), C26.21(C4⋊C4), (C2×C104).15C2, (C2×C52).119C22, (C13×M4(2)).4C2, C2.5(C13×C4⋊C4), (C2×C4).22(C2×C26), SmallGroup(416,58)

Series: Derived Chief Lower central Upper central

C1C4 — C13×C8.C4
C1C2C4C2×C4C2×C52C13×M4(2) — C13×C8.C4
C1C2C4 — C13×C8.C4
C1C52C2×C52 — C13×C8.C4

Generators and relations for C13×C8.C4
 G = < a,b,c | a13=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C2
2C26
2C8
2C8
2C104
2C104

Smallest permutation representation of C13×C8.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)
(1 147 51 138 192 22 34 55)(2 148 52 139 193 23 35 56)(3 149 40 140 194 24 36 57)(4 150 41 141 195 25 37 58)(5 151 42 142 183 26 38 59)(6 152 43 143 184 14 39 60)(7 153 44 131 185 15 27 61)(8 154 45 132 186 16 28 62)(9 155 46 133 187 17 29 63)(10 156 47 134 188 18 30 64)(11 144 48 135 189 19 31 65)(12 145 49 136 190 20 32 53)(13 146 50 137 191 21 33 54)(66 105 94 79 201 178 169 125)(67 106 95 80 202 179 157 126)(68 107 96 81 203 180 158 127)(69 108 97 82 204 181 159 128)(70 109 98 83 205 182 160 129)(71 110 99 84 206 170 161 130)(72 111 100 85 207 171 162 118)(73 112 101 86 208 172 163 119)(74 113 102 87 196 173 164 120)(75 114 103 88 197 174 165 121)(76 115 104 89 198 175 166 122)(77 116 92 90 199 176 167 123)(78 117 93 91 200 177 168 124)
(1 121 51 174 192 88 34 114)(2 122 52 175 193 89 35 115)(3 123 40 176 194 90 36 116)(4 124 41 177 195 91 37 117)(5 125 42 178 183 79 38 105)(6 126 43 179 184 80 39 106)(7 127 44 180 185 81 27 107)(8 128 45 181 186 82 28 108)(9 129 46 182 187 83 29 109)(10 130 47 170 188 84 30 110)(11 118 48 171 189 85 31 111)(12 119 49 172 190 86 32 112)(13 120 50 173 191 87 33 113)(14 95 60 67 152 157 143 202)(15 96 61 68 153 158 131 203)(16 97 62 69 154 159 132 204)(17 98 63 70 155 160 133 205)(18 99 64 71 156 161 134 206)(19 100 65 72 144 162 135 207)(20 101 53 73 145 163 136 208)(21 102 54 74 146 164 137 196)(22 103 55 75 147 165 138 197)(23 104 56 76 148 166 139 198)(24 92 57 77 149 167 140 199)(25 93 58 78 150 168 141 200)(26 94 59 66 151 169 142 201)

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

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

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

182 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F8G8H13A···13L26A···26L26M···26X52A···52X52Y···52AJ104A···104AV104AW···104CR
order1224448888888813···1326···2626···2652···5252···52104···104104···104
size112112222244441···11···12···21···12···22···24···4

182 irreducible representations

dim11111111222222
type++++-
imageC1C2C2C4C13C26C26C52D4Q8C8.C4D4×C13Q8×C13C13×C8.C4
kernelC13×C8.C4C2×C104C13×M4(2)C104C8.C4C2×C8M4(2)C8C52C2×C26C13C4C22C1
# reps112412122448114121248

Matrix representation of C13×C8.C4 in GL2(𝔽313) generated by

2340
0234
,
3080
134125
,
205311
29108
G:=sub<GL(2,GF(313))| [234,0,0,234],[308,134,0,125],[205,29,311,108] >;

C13×C8.C4 in GAP, Magma, Sage, TeX

C_{13}\times C_8.C_4
% in TeX

G:=Group("C13xC8.C4");
// GroupNames label

G:=SmallGroup(416,58);
// by ID

G=gap.SmallGroup(416,58);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,319,6243,117,88]);
// Polycyclic

G:=Group<a,b,c|a^13=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C13×C8.C4 in TeX

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