metabelian, supersoluble, monomial, A-group
Aliases: C36.6S3, C12.6D9, C6.3Dic9, C18.3Dic3, C3⋊(C9⋊C8), C9⋊(C3⋊C8), (C3×C9)⋊3C8, C4.2(C9⋊S3), (C3×C18).3C4, (C3×C36).5C2, C2.(C9⋊Dic3), C12.4(C3⋊S3), (C3×C12).16S3, C32.3(C3⋊C8), C3.(C32⋊4C8), (C3×C6).7Dic3, C6.1(C3⋊Dic3), SmallGroup(216,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C36.S3 |
Generators and relations for C36.S3
G = < a,b,c | a36=1, b3=a12, c2=a9, ab=ba, cac-1=a17, cbc-1=a24b2 >
27C8
Smallest permutation representation of C36.S3
►Regular action on 216 points
Generators in S216
(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)
(1 164 110 13 176 122 25 152 134)(2 165 111 14 177 123 26 153 135)(3 166 112 15 178 124 27 154 136)(4 167 113 16 179 125 28 155 137)(5 168 114 17 180 126 29 156 138)(6 169 115 18 145 127 30 157 139)(7 170 116 19 146 128 31 158 140)(8 171 117 20 147 129 32 159 141)(9 172 118 21 148 130 33 160 142)(10 173 119 22 149 131 34 161 143)(11 174 120 23 150 132 35 162 144)(12 175 121 24 151 133 36 163 109)(37 84 215 49 96 191 61 108 203)(38 85 216 50 97 192 62 73 204)(39 86 181 51 98 193 63 74 205)(40 87 182 52 99 194 64 75 206)(41 88 183 53 100 195 65 76 207)(42 89 184 54 101 196 66 77 208)(43 90 185 55 102 197 67 78 209)(44 91 186 56 103 198 68 79 210)(45 92 187 57 104 199 69 80 211)(46 93 188 58 105 200 70 81 212)(47 94 189 59 106 201 71 82 213)(48 95 190 60 107 202 72 83 214)
(1 195 10 204 19 213 28 186)(2 212 11 185 20 194 29 203)(3 193 12 202 21 211 30 184)(4 210 13 183 22 192 31 201)(5 191 14 200 23 209 32 182)(6 208 15 181 24 190 33 199)(7 189 16 198 25 207 34 216)(8 206 17 215 26 188 35 197)(9 187 18 196 27 205 36 214)(37 135 46 144 55 117 64 126)(38 116 47 125 56 134 65 143)(39 133 48 142 57 115 66 124)(40 114 49 123 58 132 67 141)(41 131 50 140 59 113 68 122)(42 112 51 121 60 130 69 139)(43 129 52 138 61 111 70 120)(44 110 53 119 62 128 71 137)(45 127 54 136 63 109 72 118)(73 146 82 155 91 164 100 173)(74 163 83 172 92 145 101 154)(75 180 84 153 93 162 102 171)(76 161 85 170 94 179 103 152)(77 178 86 151 95 160 104 169)(78 159 87 168 96 177 105 150)(79 176 88 149 97 158 106 167)(80 157 89 166 98 175 107 148)(81 174 90 147 99 156 108 165)
G:=sub<Sym(216)| (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), (1,164,110,13,176,122,25,152,134)(2,165,111,14,177,123,26,153,135)(3,166,112,15,178,124,27,154,136)(4,167,113,16,179,125,28,155,137)(5,168,114,17,180,126,29,156,138)(6,169,115,18,145,127,30,157,139)(7,170,116,19,146,128,31,158,140)(8,171,117,20,147,129,32,159,141)(9,172,118,21,148,130,33,160,142)(10,173,119,22,149,131,34,161,143)(11,174,120,23,150,132,35,162,144)(12,175,121,24,151,133,36,163,109)(37,84,215,49,96,191,61,108,203)(38,85,216,50,97,192,62,73,204)(39,86,181,51,98,193,63,74,205)(40,87,182,52,99,194,64,75,206)(41,88,183,53,100,195,65,76,207)(42,89,184,54,101,196,66,77,208)(43,90,185,55,102,197,67,78,209)(44,91,186,56,103,198,68,79,210)(45,92,187,57,104,199,69,80,211)(46,93,188,58,105,200,70,81,212)(47,94,189,59,106,201,71,82,213)(48,95,190,60,107,202,72,83,214), (1,195,10,204,19,213,28,186)(2,212,11,185,20,194,29,203)(3,193,12,202,21,211,30,184)(4,210,13,183,22,192,31,201)(5,191,14,200,23,209,32,182)(6,208,15,181,24,190,33,199)(7,189,16,198,25,207,34,216)(8,206,17,215,26,188,35,197)(9,187,18,196,27,205,36,214)(37,135,46,144,55,117,64,126)(38,116,47,125,56,134,65,143)(39,133,48,142,57,115,66,124)(40,114,49,123,58,132,67,141)(41,131,50,140,59,113,68,122)(42,112,51,121,60,130,69,139)(43,129,52,138,61,111,70,120)(44,110,53,119,62,128,71,137)(45,127,54,136,63,109,72,118)(73,146,82,155,91,164,100,173)(74,163,83,172,92,145,101,154)(75,180,84,153,93,162,102,171)(76,161,85,170,94,179,103,152)(77,178,86,151,95,160,104,169)(78,159,87,168,96,177,105,150)(79,176,88,149,97,158,106,167)(80,157,89,166,98,175,107,148)(81,174,90,147,99,156,108,165)>;
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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216), (1,164,110,13,176,122,25,152,134)(2,165,111,14,177,123,26,153,135)(3,166,112,15,178,124,27,154,136)(4,167,113,16,179,125,28,155,137)(5,168,114,17,180,126,29,156,138)(6,169,115,18,145,127,30,157,139)(7,170,116,19,146,128,31,158,140)(8,171,117,20,147,129,32,159,141)(9,172,118,21,148,130,33,160,142)(10,173,119,22,149,131,34,161,143)(11,174,120,23,150,132,35,162,144)(12,175,121,24,151,133,36,163,109)(37,84,215,49,96,191,61,108,203)(38,85,216,50,97,192,62,73,204)(39,86,181,51,98,193,63,74,205)(40,87,182,52,99,194,64,75,206)(41,88,183,53,100,195,65,76,207)(42,89,184,54,101,196,66,77,208)(43,90,185,55,102,197,67,78,209)(44,91,186,56,103,198,68,79,210)(45,92,187,57,104,199,69,80,211)(46,93,188,58,105,200,70,81,212)(47,94,189,59,106,201,71,82,213)(48,95,190,60,107,202,72,83,214), (1,195,10,204,19,213,28,186)(2,212,11,185,20,194,29,203)(3,193,12,202,21,211,30,184)(4,210,13,183,22,192,31,201)(5,191,14,200,23,209,32,182)(6,208,15,181,24,190,33,199)(7,189,16,198,25,207,34,216)(8,206,17,215,26,188,35,197)(9,187,18,196,27,205,36,214)(37,135,46,144,55,117,64,126)(38,116,47,125,56,134,65,143)(39,133,48,142,57,115,66,124)(40,114,49,123,58,132,67,141)(41,131,50,140,59,113,68,122)(42,112,51,121,60,130,69,139)(43,129,52,138,61,111,70,120)(44,110,53,119,62,128,71,137)(45,127,54,136,63,109,72,118)(73,146,82,155,91,164,100,173)(74,163,83,172,92,145,101,154)(75,180,84,153,93,162,102,171)(76,161,85,170,94,179,103,152)(77,178,86,151,95,160,104,169)(78,159,87,168,96,177,105,150)(79,176,88,149,97,158,106,167)(80,157,89,166,98,175,107,148)(81,174,90,147,99,156,108,165) );
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),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)], [(1,164,110,13,176,122,25,152,134),(2,165,111,14,177,123,26,153,135),(3,166,112,15,178,124,27,154,136),(4,167,113,16,179,125,28,155,137),(5,168,114,17,180,126,29,156,138),(6,169,115,18,145,127,30,157,139),(7,170,116,19,146,128,31,158,140),(8,171,117,20,147,129,32,159,141),(9,172,118,21,148,130,33,160,142),(10,173,119,22,149,131,34,161,143),(11,174,120,23,150,132,35,162,144),(12,175,121,24,151,133,36,163,109),(37,84,215,49,96,191,61,108,203),(38,85,216,50,97,192,62,73,204),(39,86,181,51,98,193,63,74,205),(40,87,182,52,99,194,64,75,206),(41,88,183,53,100,195,65,76,207),(42,89,184,54,101,196,66,77,208),(43,90,185,55,102,197,67,78,209),(44,91,186,56,103,198,68,79,210),(45,92,187,57,104,199,69,80,211),(46,93,188,58,105,200,70,81,212),(47,94,189,59,106,201,71,82,213),(48,95,190,60,107,202,72,83,214)], [(1,195,10,204,19,213,28,186),(2,212,11,185,20,194,29,203),(3,193,12,202,21,211,30,184),(4,210,13,183,22,192,31,201),(5,191,14,200,23,209,32,182),(6,208,15,181,24,190,33,199),(7,189,16,198,25,207,34,216),(8,206,17,215,26,188,35,197),(9,187,18,196,27,205,36,214),(37,135,46,144,55,117,64,126),(38,116,47,125,56,134,65,143),(39,133,48,142,57,115,66,124),(40,114,49,123,58,132,67,141),(41,131,50,140,59,113,68,122),(42,112,51,121,60,130,69,139),(43,129,52,138,61,111,70,120),(44,110,53,119,62,128,71,137),(45,127,54,136,63,109,72,118),(73,146,82,155,91,164,100,173),(74,163,83,172,92,145,101,154),(75,180,84,153,93,162,102,171),(76,161,85,170,94,179,103,152),(77,178,86,151,95,160,104,169),(78,159,87,168,96,177,105,150),(79,176,88,149,97,158,106,167),(80,157,89,166,98,175,107,148),(81,174,90,147,99,156,108,165)]])
C36.S3 is a maximal subgroup of
D9×C3⋊C8 C36.39D6 S3×C9⋊C8 D6.Dic9 D36⋊S3 D12.D9 Dic6⋊D9 C12.D18 C8×C9⋊S3 C72⋊S3 C36.69D6 C36.17D6 C36.18D6 C36.19D6 C36.20D6
C36.S3 is a maximal quotient of
C72.S3
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 9A | ··· | 9I | 12A | ··· | 12H | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 27 | 27 | 27 | 27 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | S3 | S3 | Dic3 | Dic3 | D9 | C3⋊C8 | C3⋊C8 | Dic9 | C9⋊C8 |
| kernel | C36.S3 | C3×C36 | C3×C18 | C3×C9 | C36 | C3×C12 | C18 | C3×C6 | C12 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 3 | 1 | 3 | 1 | 9 | 6 | 2 | 9 | 18 |
Matrix representation of C36.S3 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 34 | 65 |
| 0 | 0 | 8 | 26 |
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 45 | 31 |
| 0 | 0 | 42 | 3 |
| 2 | 62 | 0 | 0 |
| 60 | 71 | 0 | 0 |
| 0 | 0 | 42 | 29 |
| 0 | 0 | 71 | 31 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,34,8,0,0,65,26],[72,1,0,0,72,0,0,0,0,0,45,42,0,0,31,3],[2,60,0,0,62,71,0,0,0,0,42,71,0,0,29,31] >;
C36.S3 in GAP, Magma, Sage, TeX
C_{36}.S_3 % in TeX
G:=Group("C36.S3"); // GroupNames label
G:=SmallGroup(216,16);
// by ID
G=gap.SmallGroup(216,16);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,12,31,2115,453,1444,5189]);
// Polycyclic
G:=Group<a,b,c|a^36=1,b^3=a^12,c^2=a^9,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=a^24*b^2>;
// generators/relations
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