metabelian, supersoluble, monomial
Aliases: C6.11D36, C18.11D12, C62.126D6, (C6×C36)⋊3C2, (C2×C36)⋊2S3, (C2×C12)⋊2D9, C9⋊2(D6⋊C4), C6.15(C4×D9), C3⋊2(D18⋊C4), C18.16(C4×S3), (C6×C12).26S3, (C2×C18).38D6, (C3×C18).46D4, (C2×C6).38D18, (C3×C6).58D12, C6.23(C9⋊D4), C2.2(C36⋊S3), C6.5(C12⋊S3), C18.23(C3⋊D4), (C6×C18).40C22, C3.(C6.11D12), C32.5(D6⋊C4), C6.15(C32⋊7D4), C2.2(C6.D18), (C2×C9⋊S3)⋊2C4, C2.5(C4×C9⋊S3), (C2×C4)⋊1(C9⋊S3), C6.10(C4×C3⋊S3), (C3×C9)⋊6(C22⋊C4), (C3×C6).72(C4×S3), (C2×C9⋊Dic3)⋊3C2, C22.6(C2×C9⋊S3), (C2×C12).2(C3⋊S3), (C3×C18).26(C2×C4), (C22×C9⋊S3).2C2, (C3×C6).98(C3⋊D4), (C2×C6).32(C2×C3⋊S3), SmallGroup(432,183)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.11D36
G = < a,b,c | a18=b12=1, c2=a9, ab=ba, cac-1=a-1, cbc-1=a9b-1 >
Subgroups: 1220 in 170 conjugacy classes, 65 normal (27 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, D9, C18, C3⋊S3, C3×C6, C2×Dic3, C2×C12, C2×C12, C22×S3, C3×C9, Dic9, C36, D18, C2×C18, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, D6⋊C4, C9⋊S3, C3×C18, C2×Dic9, C2×C36, C22×D9, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C9⋊Dic3, C3×C36, C2×C9⋊S3, C2×C9⋊S3, C6×C18, D18⋊C4, C6.11D12, C2×C9⋊Dic3, C6×C36, C22×C9⋊S3, C6.11D36
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D9, C3⋊S3, C4×S3, D12, C3⋊D4, D18, C2×C3⋊S3, D6⋊C4, C9⋊S3, C4×D9, D36, C9⋊D4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C2×C9⋊S3, D18⋊C4, C6.11D12, C4×C9⋊S3, C36⋊S3, C6.D18, C6.11D36
►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)
(1 100 19 78 111 47 207 161 196 134 65 171)(2 101 20 79 112 48 208 162 197 135 66 172)(3 102 21 80 113 49 209 145 198 136 67 173)(4 103 22 81 114 50 210 146 181 137 68 174)(5 104 23 82 115 51 211 147 182 138 69 175)(6 105 24 83 116 52 212 148 183 139 70 176)(7 106 25 84 117 53 213 149 184 140 71 177)(8 107 26 85 118 54 214 150 185 141 72 178)(9 108 27 86 119 37 215 151 186 142 55 179)(10 91 28 87 120 38 216 152 187 143 56 180)(11 92 29 88 121 39 199 153 188 144 57 163)(12 93 30 89 122 40 200 154 189 127 58 164)(13 94 31 90 123 41 201 155 190 128 59 165)(14 95 32 73 124 42 202 156 191 129 60 166)(15 96 33 74 125 43 203 157 192 130 61 167)(16 97 34 75 126 44 204 158 193 131 62 168)(17 98 35 76 109 45 205 159 194 132 63 169)(18 99 36 77 110 46 206 160 195 133 64 170)
(1 38 10 47)(2 37 11 46)(3 54 12 45)(4 53 13 44)(5 52 14 43)(6 51 15 42)(7 50 16 41)(8 49 17 40)(9 48 18 39)(19 87 28 78)(20 86 29 77)(21 85 30 76)(22 84 31 75)(23 83 32 74)(24 82 33 73)(25 81 34 90)(26 80 35 89)(27 79 36 88)(55 162 64 153)(56 161 65 152)(57 160 66 151)(58 159 67 150)(59 158 68 149)(60 157 69 148)(61 156 70 147)(62 155 71 146)(63 154 72 145)(91 120 100 111)(92 119 101 110)(93 118 102 109)(94 117 103 126)(95 116 104 125)(96 115 105 124)(97 114 106 123)(98 113 107 122)(99 112 108 121)(127 185 136 194)(128 184 137 193)(129 183 138 192)(130 182 139 191)(131 181 140 190)(132 198 141 189)(133 197 142 188)(134 196 143 187)(135 195 144 186)(163 215 172 206)(164 214 173 205)(165 213 174 204)(166 212 175 203)(167 211 176 202)(168 210 177 201)(169 209 178 200)(170 208 179 199)(171 207 180 216)
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), (1,100,19,78,111,47,207,161,196,134,65,171)(2,101,20,79,112,48,208,162,197,135,66,172)(3,102,21,80,113,49,209,145,198,136,67,173)(4,103,22,81,114,50,210,146,181,137,68,174)(5,104,23,82,115,51,211,147,182,138,69,175)(6,105,24,83,116,52,212,148,183,139,70,176)(7,106,25,84,117,53,213,149,184,140,71,177)(8,107,26,85,118,54,214,150,185,141,72,178)(9,108,27,86,119,37,215,151,186,142,55,179)(10,91,28,87,120,38,216,152,187,143,56,180)(11,92,29,88,121,39,199,153,188,144,57,163)(12,93,30,89,122,40,200,154,189,127,58,164)(13,94,31,90,123,41,201,155,190,128,59,165)(14,95,32,73,124,42,202,156,191,129,60,166)(15,96,33,74,125,43,203,157,192,130,61,167)(16,97,34,75,126,44,204,158,193,131,62,168)(17,98,35,76,109,45,205,159,194,132,63,169)(18,99,36,77,110,46,206,160,195,133,64,170), (1,38,10,47)(2,37,11,46)(3,54,12,45)(4,53,13,44)(5,52,14,43)(6,51,15,42)(7,50,16,41)(8,49,17,40)(9,48,18,39)(19,87,28,78)(20,86,29,77)(21,85,30,76)(22,84,31,75)(23,83,32,74)(24,82,33,73)(25,81,34,90)(26,80,35,89)(27,79,36,88)(55,162,64,153)(56,161,65,152)(57,160,66,151)(58,159,67,150)(59,158,68,149)(60,157,69,148)(61,156,70,147)(62,155,71,146)(63,154,72,145)(91,120,100,111)(92,119,101,110)(93,118,102,109)(94,117,103,126)(95,116,104,125)(96,115,105,124)(97,114,106,123)(98,113,107,122)(99,112,108,121)(127,185,136,194)(128,184,137,193)(129,183,138,192)(130,182,139,191)(131,181,140,190)(132,198,141,189)(133,197,142,188)(134,196,143,187)(135,195,144,186)(163,215,172,206)(164,214,173,205)(165,213,174,204)(166,212,175,203)(167,211,176,202)(168,210,177,201)(169,209,178,200)(170,208,179,199)(171,207,180,216)>;
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), (1,100,19,78,111,47,207,161,196,134,65,171)(2,101,20,79,112,48,208,162,197,135,66,172)(3,102,21,80,113,49,209,145,198,136,67,173)(4,103,22,81,114,50,210,146,181,137,68,174)(5,104,23,82,115,51,211,147,182,138,69,175)(6,105,24,83,116,52,212,148,183,139,70,176)(7,106,25,84,117,53,213,149,184,140,71,177)(8,107,26,85,118,54,214,150,185,141,72,178)(9,108,27,86,119,37,215,151,186,142,55,179)(10,91,28,87,120,38,216,152,187,143,56,180)(11,92,29,88,121,39,199,153,188,144,57,163)(12,93,30,89,122,40,200,154,189,127,58,164)(13,94,31,90,123,41,201,155,190,128,59,165)(14,95,32,73,124,42,202,156,191,129,60,166)(15,96,33,74,125,43,203,157,192,130,61,167)(16,97,34,75,126,44,204,158,193,131,62,168)(17,98,35,76,109,45,205,159,194,132,63,169)(18,99,36,77,110,46,206,160,195,133,64,170), (1,38,10,47)(2,37,11,46)(3,54,12,45)(4,53,13,44)(5,52,14,43)(6,51,15,42)(7,50,16,41)(8,49,17,40)(9,48,18,39)(19,87,28,78)(20,86,29,77)(21,85,30,76)(22,84,31,75)(23,83,32,74)(24,82,33,73)(25,81,34,90)(26,80,35,89)(27,79,36,88)(55,162,64,153)(56,161,65,152)(57,160,66,151)(58,159,67,150)(59,158,68,149)(60,157,69,148)(61,156,70,147)(62,155,71,146)(63,154,72,145)(91,120,100,111)(92,119,101,110)(93,118,102,109)(94,117,103,126)(95,116,104,125)(96,115,105,124)(97,114,106,123)(98,113,107,122)(99,112,108,121)(127,185,136,194)(128,184,137,193)(129,183,138,192)(130,182,139,191)(131,181,140,190)(132,198,141,189)(133,197,142,188)(134,196,143,187)(135,195,144,186)(163,215,172,206)(164,214,173,205)(165,213,174,204)(166,212,175,203)(167,211,176,202)(168,210,177,201)(169,209,178,200)(170,208,179,199)(171,207,180,216) );
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)], [(1,100,19,78,111,47,207,161,196,134,65,171),(2,101,20,79,112,48,208,162,197,135,66,172),(3,102,21,80,113,49,209,145,198,136,67,173),(4,103,22,81,114,50,210,146,181,137,68,174),(5,104,23,82,115,51,211,147,182,138,69,175),(6,105,24,83,116,52,212,148,183,139,70,176),(7,106,25,84,117,53,213,149,184,140,71,177),(8,107,26,85,118,54,214,150,185,141,72,178),(9,108,27,86,119,37,215,151,186,142,55,179),(10,91,28,87,120,38,216,152,187,143,56,180),(11,92,29,88,121,39,199,153,188,144,57,163),(12,93,30,89,122,40,200,154,189,127,58,164),(13,94,31,90,123,41,201,155,190,128,59,165),(14,95,32,73,124,42,202,156,191,129,60,166),(15,96,33,74,125,43,203,157,192,130,61,167),(16,97,34,75,126,44,204,158,193,131,62,168),(17,98,35,76,109,45,205,159,194,132,63,169),(18,99,36,77,110,46,206,160,195,133,64,170)], [(1,38,10,47),(2,37,11,46),(3,54,12,45),(4,53,13,44),(5,52,14,43),(6,51,15,42),(7,50,16,41),(8,49,17,40),(9,48,18,39),(19,87,28,78),(20,86,29,77),(21,85,30,76),(22,84,31,75),(23,83,32,74),(24,82,33,73),(25,81,34,90),(26,80,35,89),(27,79,36,88),(55,162,64,153),(56,161,65,152),(57,160,66,151),(58,159,67,150),(59,158,68,149),(60,157,69,148),(61,156,70,147),(62,155,71,146),(63,154,72,145),(91,120,100,111),(92,119,101,110),(93,118,102,109),(94,117,103,126),(95,116,104,125),(96,115,105,124),(97,114,106,123),(98,113,107,122),(99,112,108,121),(127,185,136,194),(128,184,137,193),(129,183,138,192),(130,182,139,191),(131,181,140,190),(132,198,141,189),(133,197,142,188),(134,196,143,187),(135,195,144,186),(163,215,172,206),(164,214,173,205),(165,213,174,204),(166,212,175,203),(167,211,176,202),(168,210,177,201),(169,209,178,200),(170,208,179,199),(171,207,180,216)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6L | 9A | ··· | 9I | 12A | ··· | 12P | 18A | ··· | 18AA | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 54 | 54 | 2 | 2 | 2 | 2 | 2 | 2 | 54 | 54 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | D6 | D6 | D9 | C4×S3 | D12 | C3⋊D4 | C4×S3 | D12 | C3⋊D4 | D18 | C4×D9 | D36 | C9⋊D4 |
| kernel | C6.11D36 | C2×C9⋊Dic3 | C6×C36 | C22×C9⋊S3 | C2×C9⋊S3 | C2×C36 | C6×C12 | C3×C18 | C2×C18 | C62 | C2×C12 | C18 | C18 | C18 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 4 | 3 | 1 | 2 | 3 | 1 | 9 | 6 | 6 | 6 | 2 | 2 | 2 | 9 | 18 | 18 | 18 |
Matrix representation of C6.11D36 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 20 | 11 |
| 0 | 0 | 26 | 31 |
| 5 | 32 | 0 | 0 |
| 5 | 10 | 0 | 0 |
| 0 | 0 | 23 | 7 |
| 0 | 0 | 30 | 30 |
| 32 | 5 | 0 | 0 |
| 10 | 5 | 0 | 0 |
| 0 | 0 | 30 | 30 |
| 0 | 0 | 23 | 7 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,20,26,0,0,11,31],[5,5,0,0,32,10,0,0,0,0,23,30,0,0,7,30],[32,10,0,0,5,5,0,0,0,0,30,23,0,0,30,7] >;
C6.11D36 in GAP, Magma, Sage, TeX
C_6._{11}D_{36} % in TeX
G:=Group("C6.11D36"); // GroupNames label
G:=SmallGroup(432,183);
// by ID
G=gap.SmallGroup(432,183);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^18=b^12=1,c^2=a^9,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^9*b^-1>;
// generators/relations