Copied to
clipboard

## G = C13×M5(2)  order 416 = 25·13

### Direct product of C13 and M5(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C13×M5(2)
 Chief series C1 — C2 — C4 — C8 — C104 — C208 — C13×M5(2)
 Lower central C1 — C2 — C13×M5(2)
 Upper central C1 — C104 — C13×M5(2)

Generators and relations for C13×M5(2)
G = < a,b,c | a13=b16=c2=1, ab=ba, ac=ca, cbc=b9 >

Smallest permutation representation of C13×M5(2)
On 208 points
Generators in S208
(1 73 57 102 93 158 117 173 134 43 193 178 32)(2 74 58 103 94 159 118 174 135 44 194 179 17)(3 75 59 104 95 160 119 175 136 45 195 180 18)(4 76 60 105 96 145 120 176 137 46 196 181 19)(5 77 61 106 81 146 121 161 138 47 197 182 20)(6 78 62 107 82 147 122 162 139 48 198 183 21)(7 79 63 108 83 148 123 163 140 33 199 184 22)(8 80 64 109 84 149 124 164 141 34 200 185 23)(9 65 49 110 85 150 125 165 142 35 201 186 24)(10 66 50 111 86 151 126 166 143 36 202 187 25)(11 67 51 112 87 152 127 167 144 37 203 188 26)(12 68 52 97 88 153 128 168 129 38 204 189 27)(13 69 53 98 89 154 113 169 130 39 205 190 28)(14 70 54 99 90 155 114 170 131 40 206 191 29)(15 71 55 100 91 156 115 171 132 41 207 192 30)(16 72 56 101 92 157 116 172 133 42 208 177 31)
(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)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)(66 74)(68 76)(70 78)(72 80)(82 90)(84 92)(86 94)(88 96)(97 105)(99 107)(101 109)(103 111)(114 122)(116 124)(118 126)(120 128)(129 137)(131 139)(133 141)(135 143)(145 153)(147 155)(149 157)(151 159)(162 170)(164 172)(166 174)(168 176)(177 185)(179 187)(181 189)(183 191)(194 202)(196 204)(198 206)(200 208)

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

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

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

260 conjugacy classes

 class 1 2A 2B 4A 4B 4C 8A 8B 8C 8D 8E 8F 13A ··· 13L 16A ··· 16H 26A ··· 26L 26M ··· 26X 52A ··· 52X 52Y ··· 52AJ 104A ··· 104AV 104AW ··· 104BT 208A ··· 208CR order 1 2 2 4 4 4 8 8 8 8 8 8 13 ··· 13 16 ··· 16 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 104 ··· 104 208 ··· 208 size 1 1 2 1 1 2 1 1 1 1 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

260 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C4 C4 C8 C8 C13 C26 C26 C52 C52 C104 C104 M5(2) C13×M5(2) kernel C13×M5(2) C208 C2×C104 C104 C2×C52 C52 C2×C26 M5(2) C16 C2×C8 C8 C2×C4 C4 C22 C13 C1 # reps 1 2 1 2 2 4 4 12 24 12 24 24 48 48 4 48

Matrix representation of C13×M5(2) in GL2(𝔽1249) generated by

 933 0 0 933
,
 1022 1247 354 227
,
 1 0 1022 1248
G:=sub<GL(2,GF(1249))| [933,0,0,933],[1022,354,1247,227],[1,1022,0,1248] >;

C13×M5(2) in GAP, Magma, Sage, TeX

C_{13}\times M_5(2)
% in TeX

G:=Group("C13xM5(2)");
// GroupNames label

G:=SmallGroup(416,60);
// by ID

G=gap.SmallGroup(416,60);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,312,2521,69,88]);
// Polycyclic

G:=Group<a,b,c|a^13=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations

Export

׿
×
𝔽