metabelian, supersoluble, monomial
Aliases: C36.27D6, C12.27D18, C62.76D6, (C3×D4)⋊3D9, (D4×C9)⋊3S3, D4⋊2(C9⋊S3), (C2×C6).6D18, (C2×C18).6D6, C9⋊5(D4⋊2S3), C3⋊5(D4⋊2D9), C12.D9⋊9C2, (C3×C12).106D6, C6.44(C22×D9), C6.D18⋊6C2, (C3×C18).53C23, (C6×C18).20C22, C18.44(C22×S3), (C3×C36).33C22, C3.(C12.D6), (D4×C32).14S3, C9⋊Dic3.16C22, C32.7(D4⋊2S3), (D4×C3×C9)⋊6C2, (C4×C9⋊S3)⋊5C2, C4.5(C2×C9⋊S3), C12.6(C2×C3⋊S3), (C3×C9)⋊16(C4○D4), (C2×C9⋊Dic3)⋊9C2, C22.1(C2×C9⋊S3), C2.7(C22×C9⋊S3), (C3×D4).4(C3⋊S3), C6.33(C22×C3⋊S3), (C2×C9⋊S3).13C22, (C3×C6).167(C22×S3), (C2×C6).5(C2×C3⋊S3), SmallGroup(432,389)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.27D6
G = < a,b,c | a36=1, b6=a24, c2=a18, bab-1=a19, cac-1=a-1, cbc-1=a30b5 >
Subgroups: 1088 in 200 conjugacy classes, 71 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, D4, Q8, C9, C32, Dic3, C12, C12, D6, C2×C6, C4○D4, D9, C18, C18, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C3×C9, Dic9, C36, D18, C2×C18, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, D4⋊2S3, C9⋊S3, C3×C18, C3×C18, Dic18, C4×D9, C2×Dic9, C9⋊D4, D4×C9, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, C9⋊Dic3, C9⋊Dic3, C3×C36, C2×C9⋊S3, C6×C18, D4⋊2D9, C12.D6, C12.D9, C4×C9⋊S3, C2×C9⋊Dic3, C6.D18, D4×C3×C9, C36.27D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C3⋊S3, C22×S3, D18, C2×C3⋊S3, D4⋊2S3, C9⋊S3, C22×D9, C22×C3⋊S3, C2×C9⋊S3, D4⋊2D9, C12.D6, C22×C9⋊S3, C36.27D6
►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 39 121 13 51 133 25 63 109)(2 58 122 32 52 116 26 46 110 20 40 140 14 70 134 8 64 128)(3 41 123 15 53 135 27 65 111)(4 60 124 34 54 118 28 48 112 22 42 142 16 72 136 10 66 130)(5 43 125 17 55 137 29 67 113)(6 62 126 36 56 120 30 50 114 24 44 144 18 38 138 12 68 132)(7 45 127 19 57 139 31 69 115)(9 47 129 21 59 141 33 71 117)(11 49 131 23 61 143 35 37 119)(73 174 189 103 168 183 97 162 213 91 156 207 85 150 201 79 180 195)(74 157 190 86 169 202 98 145 214)(75 176 191 105 170 185 99 164 215 93 158 209 87 152 203 81 146 197)(76 159 192 88 171 204 100 147 216)(77 178 193 107 172 187 101 166 181 95 160 211 89 154 205 83 148 199)(78 161 194 90 173 206 102 149 182)(80 163 196 92 175 208 104 151 184)(82 165 198 94 177 210 106 153 186)(84 167 200 96 179 212 108 155 188)
(1 83 19 101)(2 82 20 100)(3 81 21 99)(4 80 22 98)(5 79 23 97)(6 78 24 96)(7 77 25 95)(8 76 26 94)(9 75 27 93)(10 74 28 92)(11 73 29 91)(12 108 30 90)(13 107 31 89)(14 106 32 88)(15 105 33 87)(16 104 34 86)(17 103 35 85)(18 102 36 84)(37 189 55 207)(38 188 56 206)(39 187 57 205)(40 186 58 204)(41 185 59 203)(42 184 60 202)(43 183 61 201)(44 182 62 200)(45 181 63 199)(46 216 64 198)(47 215 65 197)(48 214 66 196)(49 213 67 195)(50 212 68 194)(51 211 69 193)(52 210 70 192)(53 209 71 191)(54 208 72 190)(109 166 127 148)(110 165 128 147)(111 164 129 146)(112 163 130 145)(113 162 131 180)(114 161 132 179)(115 160 133 178)(116 159 134 177)(117 158 135 176)(118 157 136 175)(119 156 137 174)(120 155 138 173)(121 154 139 172)(122 153 140 171)(123 152 141 170)(124 151 142 169)(125 150 143 168)(126 149 144 167)
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), (1,39,121,13,51,133,25,63,109)(2,58,122,32,52,116,26,46,110,20,40,140,14,70,134,8,64,128)(3,41,123,15,53,135,27,65,111)(4,60,124,34,54,118,28,48,112,22,42,142,16,72,136,10,66,130)(5,43,125,17,55,137,29,67,113)(6,62,126,36,56,120,30,50,114,24,44,144,18,38,138,12,68,132)(7,45,127,19,57,139,31,69,115)(9,47,129,21,59,141,33,71,117)(11,49,131,23,61,143,35,37,119)(73,174,189,103,168,183,97,162,213,91,156,207,85,150,201,79,180,195)(74,157,190,86,169,202,98,145,214)(75,176,191,105,170,185,99,164,215,93,158,209,87,152,203,81,146,197)(76,159,192,88,171,204,100,147,216)(77,178,193,107,172,187,101,166,181,95,160,211,89,154,205,83,148,199)(78,161,194,90,173,206,102,149,182)(80,163,196,92,175,208,104,151,184)(82,165,198,94,177,210,106,153,186)(84,167,200,96,179,212,108,155,188), (1,83,19,101)(2,82,20,100)(3,81,21,99)(4,80,22,98)(5,79,23,97)(6,78,24,96)(7,77,25,95)(8,76,26,94)(9,75,27,93)(10,74,28,92)(11,73,29,91)(12,108,30,90)(13,107,31,89)(14,106,32,88)(15,105,33,87)(16,104,34,86)(17,103,35,85)(18,102,36,84)(37,189,55,207)(38,188,56,206)(39,187,57,205)(40,186,58,204)(41,185,59,203)(42,184,60,202)(43,183,61,201)(44,182,62,200)(45,181,63,199)(46,216,64,198)(47,215,65,197)(48,214,66,196)(49,213,67,195)(50,212,68,194)(51,211,69,193)(52,210,70,192)(53,209,71,191)(54,208,72,190)(109,166,127,148)(110,165,128,147)(111,164,129,146)(112,163,130,145)(113,162,131,180)(114,161,132,179)(115,160,133,178)(116,159,134,177)(117,158,135,176)(118,157,136,175)(119,156,137,174)(120,155,138,173)(121,154,139,172)(122,153,140,171)(123,152,141,170)(124,151,142,169)(125,150,143,168)(126,149,144,167)>;
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), (1,39,121,13,51,133,25,63,109)(2,58,122,32,52,116,26,46,110,20,40,140,14,70,134,8,64,128)(3,41,123,15,53,135,27,65,111)(4,60,124,34,54,118,28,48,112,22,42,142,16,72,136,10,66,130)(5,43,125,17,55,137,29,67,113)(6,62,126,36,56,120,30,50,114,24,44,144,18,38,138,12,68,132)(7,45,127,19,57,139,31,69,115)(9,47,129,21,59,141,33,71,117)(11,49,131,23,61,143,35,37,119)(73,174,189,103,168,183,97,162,213,91,156,207,85,150,201,79,180,195)(74,157,190,86,169,202,98,145,214)(75,176,191,105,170,185,99,164,215,93,158,209,87,152,203,81,146,197)(76,159,192,88,171,204,100,147,216)(77,178,193,107,172,187,101,166,181,95,160,211,89,154,205,83,148,199)(78,161,194,90,173,206,102,149,182)(80,163,196,92,175,208,104,151,184)(82,165,198,94,177,210,106,153,186)(84,167,200,96,179,212,108,155,188), (1,83,19,101)(2,82,20,100)(3,81,21,99)(4,80,22,98)(5,79,23,97)(6,78,24,96)(7,77,25,95)(8,76,26,94)(9,75,27,93)(10,74,28,92)(11,73,29,91)(12,108,30,90)(13,107,31,89)(14,106,32,88)(15,105,33,87)(16,104,34,86)(17,103,35,85)(18,102,36,84)(37,189,55,207)(38,188,56,206)(39,187,57,205)(40,186,58,204)(41,185,59,203)(42,184,60,202)(43,183,61,201)(44,182,62,200)(45,181,63,199)(46,216,64,198)(47,215,65,197)(48,214,66,196)(49,213,67,195)(50,212,68,194)(51,211,69,193)(52,210,70,192)(53,209,71,191)(54,208,72,190)(109,166,127,148)(110,165,128,147)(111,164,129,146)(112,163,130,145)(113,162,131,180)(114,161,132,179)(115,160,133,178)(116,159,134,177)(117,158,135,176)(118,157,136,175)(119,156,137,174)(120,155,138,173)(121,154,139,172)(122,153,140,171)(123,152,141,170)(124,151,142,169)(125,150,143,168)(126,149,144,167) );
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)], [(1,39,121,13,51,133,25,63,109),(2,58,122,32,52,116,26,46,110,20,40,140,14,70,134,8,64,128),(3,41,123,15,53,135,27,65,111),(4,60,124,34,54,118,28,48,112,22,42,142,16,72,136,10,66,130),(5,43,125,17,55,137,29,67,113),(6,62,126,36,56,120,30,50,114,24,44,144,18,38,138,12,68,132),(7,45,127,19,57,139,31,69,115),(9,47,129,21,59,141,33,71,117),(11,49,131,23,61,143,35,37,119),(73,174,189,103,168,183,97,162,213,91,156,207,85,150,201,79,180,195),(74,157,190,86,169,202,98,145,214),(75,176,191,105,170,185,99,164,215,93,158,209,87,152,203,81,146,197),(76,159,192,88,171,204,100,147,216),(77,178,193,107,172,187,101,166,181,95,160,211,89,154,205,83,148,199),(78,161,194,90,173,206,102,149,182),(80,163,196,92,175,208,104,151,184),(82,165,198,94,177,210,106,153,186),(84,167,200,96,179,212,108,155,188)], [(1,83,19,101),(2,82,20,100),(3,81,21,99),(4,80,22,98),(5,79,23,97),(6,78,24,96),(7,77,25,95),(8,76,26,94),(9,75,27,93),(10,74,28,92),(11,73,29,91),(12,108,30,90),(13,107,31,89),(14,106,32,88),(15,105,33,87),(16,104,34,86),(17,103,35,85),(18,102,36,84),(37,189,55,207),(38,188,56,206),(39,187,57,205),(40,186,58,204),(41,185,59,203),(42,184,60,202),(43,183,61,201),(44,182,62,200),(45,181,63,199),(46,216,64,198),(47,215,65,197),(48,214,66,196),(49,213,67,195),(50,212,68,194),(51,211,69,193),(52,210,70,192),(53,209,71,191),(54,208,72,190),(109,166,127,148),(110,165,128,147),(111,164,129,146),(112,163,130,145),(113,162,131,180),(114,161,132,179),(115,160,133,178),(116,159,134,177),(117,158,135,176),(118,157,136,175),(119,156,137,174),(120,155,138,173),(121,154,139,172),(122,153,140,171),(123,152,141,170),(124,151,142,169),(125,150,143,168),(126,149,144,167)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 9A | ··· | 9I | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18AA | 36A | ··· | 36I |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 54 | 2 | 2 | 2 | 2 | 2 | 27 | 27 | 54 | 54 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | C4○D4 | D9 | D18 | D18 | D4⋊2S3 | D4⋊2S3 | D4⋊2D9 |
| kernel | C36.27D6 | C12.D9 | C4×C9⋊S3 | C2×C9⋊Dic3 | C6.D18 | D4×C3×C9 | D4×C9 | D4×C32 | C36 | C2×C18 | C3×C12 | C62 | C3×C9 | C3×D4 | C12 | C2×C6 | C9 | C32 | C3 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 3 | 1 | 3 | 6 | 1 | 2 | 2 | 9 | 9 | 18 | 3 | 1 | 9 |
Matrix representation of C36.27D6 ►in GL6(𝔽37)
| 11 | 31 | 0 | 0 | 0 | 0 |
| 6 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 31 | 0 | 0 |
| 0 | 0 | 6 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 3 |
| 0 | 0 | 0 | 0 | 24 | 1 |
| 31 | 20 | 0 | 0 | 0 | 0 |
| 17 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 11 | 0 | 0 |
| 0 | 0 | 26 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 13 | 36 |
| 4 | 29 | 0 | 0 | 0 | 0 |
| 25 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 8 | 0 | 0 |
| 0 | 0 | 33 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 18 |
| 0 | 0 | 0 | 0 | 0 | 6 |
G:=sub<GL(6,GF(37))| [11,6,0,0,0,0,31,17,0,0,0,0,0,0,11,6,0,0,0,0,31,17,0,0,0,0,0,0,36,24,0,0,0,0,3,1],[31,17,0,0,0,0,20,11,0,0,0,0,0,0,17,26,0,0,0,0,11,6,0,0,0,0,0,0,1,13,0,0,0,0,0,36],[4,25,0,0,0,0,29,33,0,0,0,0,0,0,12,33,0,0,0,0,8,25,0,0,0,0,0,0,31,0,0,0,0,0,18,6] >;
C36.27D6 in GAP, Magma, Sage, TeX
C_{36}._{27}D_6 % in TeX
G:=Group("C36.27D6"); // GroupNames label
G:=SmallGroup(432,389);
// by ID
G=gap.SmallGroup(432,389);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^36=1,b^6=a^24,c^2=a^18,b*a*b^-1=a^19,c*a*c^-1=a^-1,c*b*c^-1=a^30*b^5>;
// generators/relations