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G = C7⋊C36order 252 = 22·32·7

The semidirect product of C7 and C36 acting via C36/C6=C6

metacyclic, supersoluble, monomial, Z-group

Aliases: C7⋊C36, Dic7⋊C9, C14.C18, C21.C12, C6.2F7, C42.2C6, C7⋊C9⋊C4, C3.(C7⋊C12), C2.(C7⋊C18), (C3×Dic7).C3, (C2×C7⋊C9).C2, SmallGroup(252,1)

Series: Derived Chief Lower central Upper central

C1C7 — C7⋊C36
C1C7C21C42C2×C7⋊C9 — C7⋊C36
C7 — C7⋊C36
C1C6

Generators and relations for C7⋊C36
 G = < a,b | a7=b36=1, bab-1=a5 >

7C4
7C9
7C12
7C18
7C36

Smallest permutation representation of C7⋊C36
Regular action on 252 points
Generators in S252
(1 248 134 192 49 164 105)(2 165 193 249 106 50 135)(3 51 250 166 136 107 194)(4 108 167 52 195 137 251)(5 138 53 73 252 196 168)(6 197 74 139 169 217 54)(7 218 140 198 55 170 75)(8 171 199 219 76 56 141)(9 57 220 172 142 77 200)(10 78 173 58 201 143 221)(11 144 59 79 222 202 174)(12 203 80 109 175 223 60)(13 224 110 204 61 176 81)(14 177 205 225 82 62 111)(15 63 226 178 112 83 206)(16 84 179 64 207 113 227)(17 114 65 85 228 208 180)(18 209 86 115 145 229 66)(19 230 116 210 67 146 87)(20 147 211 231 88 68 117)(21 69 232 148 118 89 212)(22 90 149 70 213 119 233)(23 120 71 91 234 214 150)(24 215 92 121 151 235 72)(25 236 122 216 37 152 93)(26 153 181 237 94 38 123)(27 39 238 154 124 95 182)(28 96 155 40 183 125 239)(29 126 41 97 240 184 156)(30 185 98 127 157 241 42)(31 242 128 186 43 158 99)(32 159 187 243 100 44 129)(33 45 244 160 130 101 188)(34 102 161 46 189 131 245)(35 132 47 103 246 190 162)(36 191 104 133 163 247 48)
(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)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252)

G:=sub<Sym(252)| (1,248,134,192,49,164,105)(2,165,193,249,106,50,135)(3,51,250,166,136,107,194)(4,108,167,52,195,137,251)(5,138,53,73,252,196,168)(6,197,74,139,169,217,54)(7,218,140,198,55,170,75)(8,171,199,219,76,56,141)(9,57,220,172,142,77,200)(10,78,173,58,201,143,221)(11,144,59,79,222,202,174)(12,203,80,109,175,223,60)(13,224,110,204,61,176,81)(14,177,205,225,82,62,111)(15,63,226,178,112,83,206)(16,84,179,64,207,113,227)(17,114,65,85,228,208,180)(18,209,86,115,145,229,66)(19,230,116,210,67,146,87)(20,147,211,231,88,68,117)(21,69,232,148,118,89,212)(22,90,149,70,213,119,233)(23,120,71,91,234,214,150)(24,215,92,121,151,235,72)(25,236,122,216,37,152,93)(26,153,181,237,94,38,123)(27,39,238,154,124,95,182)(28,96,155,40,183,125,239)(29,126,41,97,240,184,156)(30,185,98,127,157,241,42)(31,242,128,186,43,158,99)(32,159,187,243,100,44,129)(33,45,244,160,130,101,188)(34,102,161,46,189,131,245)(35,132,47,103,246,190,162)(36,191,104,133,163,247,48), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)>;

G:=Group( (1,248,134,192,49,164,105)(2,165,193,249,106,50,135)(3,51,250,166,136,107,194)(4,108,167,52,195,137,251)(5,138,53,73,252,196,168)(6,197,74,139,169,217,54)(7,218,140,198,55,170,75)(8,171,199,219,76,56,141)(9,57,220,172,142,77,200)(10,78,173,58,201,143,221)(11,144,59,79,222,202,174)(12,203,80,109,175,223,60)(13,224,110,204,61,176,81)(14,177,205,225,82,62,111)(15,63,226,178,112,83,206)(16,84,179,64,207,113,227)(17,114,65,85,228,208,180)(18,209,86,115,145,229,66)(19,230,116,210,67,146,87)(20,147,211,231,88,68,117)(21,69,232,148,118,89,212)(22,90,149,70,213,119,233)(23,120,71,91,234,214,150)(24,215,92,121,151,235,72)(25,236,122,216,37,152,93)(26,153,181,237,94,38,123)(27,39,238,154,124,95,182)(28,96,155,40,183,125,239)(29,126,41,97,240,184,156)(30,185,98,127,157,241,42)(31,242,128,186,43,158,99)(32,159,187,243,100,44,129)(33,45,244,160,130,101,188)(34,102,161,46,189,131,245)(35,132,47,103,246,190,162)(36,191,104,133,163,247,48), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252) );

G=PermutationGroup([(1,248,134,192,49,164,105),(2,165,193,249,106,50,135),(3,51,250,166,136,107,194),(4,108,167,52,195,137,251),(5,138,53,73,252,196,168),(6,197,74,139,169,217,54),(7,218,140,198,55,170,75),(8,171,199,219,76,56,141),(9,57,220,172,142,77,200),(10,78,173,58,201,143,221),(11,144,59,79,222,202,174),(12,203,80,109,175,223,60),(13,224,110,204,61,176,81),(14,177,205,225,82,62,111),(15,63,226,178,112,83,206),(16,84,179,64,207,113,227),(17,114,65,85,228,208,180),(18,209,86,115,145,229,66),(19,230,116,210,67,146,87),(20,147,211,231,88,68,117),(21,69,232,148,118,89,212),(22,90,149,70,213,119,233),(23,120,71,91,234,214,150),(24,215,92,121,151,235,72),(25,236,122,216,37,152,93),(26,153,181,237,94,38,123),(27,39,238,154,124,95,182),(28,96,155,40,183,125,239),(29,126,41,97,240,184,156),(30,185,98,127,157,241,42),(31,242,128,186,43,158,99),(32,159,187,243,100,44,129),(33,45,244,160,130,101,188),(34,102,161,46,189,131,245),(35,132,47,103,246,190,162),(36,191,104,133,163,247,48)], [(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),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)])

42 conjugacy classes

class 1  2 3A3B4A4B6A6B 7 9A···9F12A12B12C12D 14 18A···18F21A21B36A···36L42A42B
order1233446679···9121212121418···18212136···364242
size1111771167···7777767···7667···766

42 irreducible representations

dim1111111116666
type+++-
imageC1C2C3C4C6C9C12C18C36F7C7⋊C12C7⋊C18C7⋊C36
kernelC7⋊C36C2×C7⋊C9C3×Dic7C7⋊C9C42Dic7C21C14C7C6C3C2C1
# reps11222646121122

Matrix representation of C7⋊C36 in GL7(𝔽757)

1000000
075610000
075601000
075600100
075600010
075600001
075600000
,
87000000
093353619435629504
071231491182267597
0584535129275620459
0222628482137298331
0575490160945726
0253362664404138322

G:=sub<GL(7,GF(757))| [1,0,0,0,0,0,0,0,756,756,756,756,756,756,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[87,0,0,0,0,0,0,0,93,712,584,222,575,253,0,353,31,535,628,490,362,0,619,491,129,482,160,664,0,435,182,275,137,9,404,0,629,267,620,298,45,138,0,504,597,459,331,726,322] >;

C7⋊C36 in GAP, Magma, Sage, TeX

C_7\rtimes C_{36}
% in TeX

G:=Group("C7:C36");
// GroupNames label

G:=SmallGroup(252,1);
// by ID

G=gap.SmallGroup(252,1);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-7,30,66,5404,1809]);
// Polycyclic

G:=Group<a,b|a^7=b^36=1,b*a*b^-1=a^5>;
// generators/relations

Export

Subgroup lattice of C7⋊C36 in TeX

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