metabelian, supersoluble, monomial
Aliases: C36.18D6, C12.18D18, (C3×C9)⋊9D8, D4⋊(C9⋊S3), (C3×D4)⋊1D9, (D4×C9)⋊1S3, C3⋊3(D4⋊D9), C9⋊3(D4⋊S3), C36⋊S3⋊6C2, (C3×C12).86D6, (C3×C18).48D4, C3.(C32⋊7D8), C36.S3⋊5C2, C6.25(C9⋊D4), C18.25(C3⋊D4), (C3×C36).21C22, C32.5(D4⋊S3), (D4×C32).10S3, C6.17(C32⋊7D4), C2.5(C6.D18), (D4×C3×C9)⋊2C2, C4.2(C2×C9⋊S3), C12.2(C2×C3⋊S3), (C3×D4).2(C3⋊S3), (C3×C6).100(C3⋊D4), SmallGroup(432,191)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.18D6
G = < a,b,c | a36=b6=1, c2=a27, bab-1=a19, cac-1=a17, cbc-1=a27b-1 >
Subgroups: 784 in 110 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C6, C8, D4, D4, C9, C32, C12, C12, D6, C2×C6, D8, D9, C18, C18, C3⋊S3, C3×C6, C3×C6, C3⋊C8, D12, C3×D4, C3×D4, C3×C9, C36, D18, C2×C18, C3×C12, C2×C3⋊S3, C62, D4⋊S3, C9⋊S3, C3×C18, C3×C18, C9⋊C8, D36, D4×C9, C32⋊4C8, C12⋊S3, D4×C32, C3×C36, C2×C9⋊S3, C6×C18, D4⋊D9, C32⋊7D8, C36.S3, C36⋊S3, D4×C3×C9, C36.18D6
Quotients: C1, C2, C22, S3, D4, D6, D8, D9, C3⋊S3, C3⋊D4, D18, C2×C3⋊S3, D4⋊S3, C9⋊S3, C9⋊D4, C32⋊7D4, C2×C9⋊S3, D4⋊D9, C32⋊7D8, C6.D18, C36.18D6
►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)
(1 171 76)(2 154 77 20 172 95)(3 173 78)(4 156 79 22 174 97)(5 175 80)(6 158 81 24 176 99)(7 177 82)(8 160 83 26 178 101)(9 179 84)(10 162 85 28 180 103)(11 145 86)(12 164 87 30 146 105)(13 147 88)(14 166 89 32 148 107)(15 149 90)(16 168 91 34 150 73)(17 151 92)(18 170 93 36 152 75)(19 153 94)(21 155 96)(23 157 98)(25 159 100)(27 161 102)(29 163 104)(31 165 106)(33 167 108)(35 169 74)(37 134 204 64 143 195)(38 117 205 47 144 214)(39 136 206 66 109 197)(40 119 207 49 110 216)(41 138 208 68 111 199)(42 121 209 51 112 182)(43 140 210 70 113 201)(44 123 211 53 114 184)(45 142 212 72 115 203)(46 125 213 55 116 186)(48 127 215 57 118 188)(50 129 181 59 120 190)(52 131 183 61 122 192)(54 133 185 63 124 194)(56 135 187 65 126 196)(58 137 189 67 128 198)(60 139 191 69 130 200)(62 141 193 71 132 202)
(1 191 28 182 19 209 10 200)(2 208 29 199 20 190 11 181)(3 189 30 216 21 207 12 198)(4 206 31 197 22 188 13 215)(5 187 32 214 23 205 14 196)(6 204 33 195 24 186 15 213)(7 185 34 212 25 203 16 194)(8 202 35 193 26 184 17 211)(9 183 36 210 27 201 18 192)(37 108 64 99 55 90 46 81)(38 89 65 80 56 107 47 98)(39 106 66 97 57 88 48 79)(40 87 67 78 58 105 49 96)(41 104 68 95 59 86 50 77)(42 85 69 76 60 103 51 94)(43 102 70 93 61 84 52 75)(44 83 71 74 62 101 53 92)(45 100 72 91 63 82 54 73)(109 165 136 156 127 147 118 174)(110 146 137 173 128 164 119 155)(111 163 138 154 129 145 120 172)(112 180 139 171 130 162 121 153)(113 161 140 152 131 179 122 170)(114 178 141 169 132 160 123 151)(115 159 142 150 133 177 124 168)(116 176 143 167 134 158 125 149)(117 157 144 148 135 175 126 166)
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), (1,171,76)(2,154,77,20,172,95)(3,173,78)(4,156,79,22,174,97)(5,175,80)(6,158,81,24,176,99)(7,177,82)(8,160,83,26,178,101)(9,179,84)(10,162,85,28,180,103)(11,145,86)(12,164,87,30,146,105)(13,147,88)(14,166,89,32,148,107)(15,149,90)(16,168,91,34,150,73)(17,151,92)(18,170,93,36,152,75)(19,153,94)(21,155,96)(23,157,98)(25,159,100)(27,161,102)(29,163,104)(31,165,106)(33,167,108)(35,169,74)(37,134,204,64,143,195)(38,117,205,47,144,214)(39,136,206,66,109,197)(40,119,207,49,110,216)(41,138,208,68,111,199)(42,121,209,51,112,182)(43,140,210,70,113,201)(44,123,211,53,114,184)(45,142,212,72,115,203)(46,125,213,55,116,186)(48,127,215,57,118,188)(50,129,181,59,120,190)(52,131,183,61,122,192)(54,133,185,63,124,194)(56,135,187,65,126,196)(58,137,189,67,128,198)(60,139,191,69,130,200)(62,141,193,71,132,202), (1,191,28,182,19,209,10,200)(2,208,29,199,20,190,11,181)(3,189,30,216,21,207,12,198)(4,206,31,197,22,188,13,215)(5,187,32,214,23,205,14,196)(6,204,33,195,24,186,15,213)(7,185,34,212,25,203,16,194)(8,202,35,193,26,184,17,211)(9,183,36,210,27,201,18,192)(37,108,64,99,55,90,46,81)(38,89,65,80,56,107,47,98)(39,106,66,97,57,88,48,79)(40,87,67,78,58,105,49,96)(41,104,68,95,59,86,50,77)(42,85,69,76,60,103,51,94)(43,102,70,93,61,84,52,75)(44,83,71,74,62,101,53,92)(45,100,72,91,63,82,54,73)(109,165,136,156,127,147,118,174)(110,146,137,173,128,164,119,155)(111,163,138,154,129,145,120,172)(112,180,139,171,130,162,121,153)(113,161,140,152,131,179,122,170)(114,178,141,169,132,160,123,151)(115,159,142,150,133,177,124,168)(116,176,143,167,134,158,125,149)(117,157,144,148,135,175,126,166)>;
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), (1,171,76)(2,154,77,20,172,95)(3,173,78)(4,156,79,22,174,97)(5,175,80)(6,158,81,24,176,99)(7,177,82)(8,160,83,26,178,101)(9,179,84)(10,162,85,28,180,103)(11,145,86)(12,164,87,30,146,105)(13,147,88)(14,166,89,32,148,107)(15,149,90)(16,168,91,34,150,73)(17,151,92)(18,170,93,36,152,75)(19,153,94)(21,155,96)(23,157,98)(25,159,100)(27,161,102)(29,163,104)(31,165,106)(33,167,108)(35,169,74)(37,134,204,64,143,195)(38,117,205,47,144,214)(39,136,206,66,109,197)(40,119,207,49,110,216)(41,138,208,68,111,199)(42,121,209,51,112,182)(43,140,210,70,113,201)(44,123,211,53,114,184)(45,142,212,72,115,203)(46,125,213,55,116,186)(48,127,215,57,118,188)(50,129,181,59,120,190)(52,131,183,61,122,192)(54,133,185,63,124,194)(56,135,187,65,126,196)(58,137,189,67,128,198)(60,139,191,69,130,200)(62,141,193,71,132,202), (1,191,28,182,19,209,10,200)(2,208,29,199,20,190,11,181)(3,189,30,216,21,207,12,198)(4,206,31,197,22,188,13,215)(5,187,32,214,23,205,14,196)(6,204,33,195,24,186,15,213)(7,185,34,212,25,203,16,194)(8,202,35,193,26,184,17,211)(9,183,36,210,27,201,18,192)(37,108,64,99,55,90,46,81)(38,89,65,80,56,107,47,98)(39,106,66,97,57,88,48,79)(40,87,67,78,58,105,49,96)(41,104,68,95,59,86,50,77)(42,85,69,76,60,103,51,94)(43,102,70,93,61,84,52,75)(44,83,71,74,62,101,53,92)(45,100,72,91,63,82,54,73)(109,165,136,156,127,147,118,174)(110,146,137,173,128,164,119,155)(111,163,138,154,129,145,120,172)(112,180,139,171,130,162,121,153)(113,161,140,152,131,179,122,170)(114,178,141,169,132,160,123,151)(115,159,142,150,133,177,124,168)(116,176,143,167,134,158,125,149)(117,157,144,148,135,175,126,166) );
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)], [(1,171,76),(2,154,77,20,172,95),(3,173,78),(4,156,79,22,174,97),(5,175,80),(6,158,81,24,176,99),(7,177,82),(8,160,83,26,178,101),(9,179,84),(10,162,85,28,180,103),(11,145,86),(12,164,87,30,146,105),(13,147,88),(14,166,89,32,148,107),(15,149,90),(16,168,91,34,150,73),(17,151,92),(18,170,93,36,152,75),(19,153,94),(21,155,96),(23,157,98),(25,159,100),(27,161,102),(29,163,104),(31,165,106),(33,167,108),(35,169,74),(37,134,204,64,143,195),(38,117,205,47,144,214),(39,136,206,66,109,197),(40,119,207,49,110,216),(41,138,208,68,111,199),(42,121,209,51,112,182),(43,140,210,70,113,201),(44,123,211,53,114,184),(45,142,212,72,115,203),(46,125,213,55,116,186),(48,127,215,57,118,188),(50,129,181,59,120,190),(52,131,183,61,122,192),(54,133,185,63,124,194),(56,135,187,65,126,196),(58,137,189,67,128,198),(60,139,191,69,130,200),(62,141,193,71,132,202)], [(1,191,28,182,19,209,10,200),(2,208,29,199,20,190,11,181),(3,189,30,216,21,207,12,198),(4,206,31,197,22,188,13,215),(5,187,32,214,23,205,14,196),(6,204,33,195,24,186,15,213),(7,185,34,212,25,203,16,194),(8,202,35,193,26,184,17,211),(9,183,36,210,27,201,18,192),(37,108,64,99,55,90,46,81),(38,89,65,80,56,107,47,98),(39,106,66,97,57,88,48,79),(40,87,67,78,58,105,49,96),(41,104,68,95,59,86,50,77),(42,85,69,76,60,103,51,94),(43,102,70,93,61,84,52,75),(44,83,71,74,62,101,53,92),(45,100,72,91,63,82,54,73),(109,165,136,156,127,147,118,174),(110,146,137,173,128,164,119,155),(111,163,138,154,129,145,120,172),(112,180,139,171,130,162,121,153),(113,161,140,152,131,179,122,170),(114,178,141,169,132,160,123,151),(115,159,142,150,133,177,124,168),(116,176,143,167,134,158,125,149),(117,157,144,148,135,175,126,166)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 8A | 8B | 9A | ··· | 9I | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18AA | 36A | ··· | 36I |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 4 | 108 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 54 | 54 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D8 | D9 | C3⋊D4 | C3⋊D4 | D18 | C9⋊D4 | D4⋊S3 | D4⋊S3 | D4⋊D9 |
| kernel | C36.18D6 | C36.S3 | C36⋊S3 | D4×C3×C9 | D4×C9 | D4×C32 | C3×C18 | C36 | C3×C12 | C3×C9 | C3×D4 | C18 | C3×C6 | C12 | C6 | C9 | C32 | C3 |
| # reps | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 3 | 1 | 2 | 9 | 6 | 2 | 9 | 18 | 3 | 1 | 9 |
Matrix representation of C36.18D6 ►in GL6(𝔽73)
| 3 | 31 | 0 | 0 | 0 | 0 |
| 42 | 45 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 1 | 65 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 30 | 65 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 55 | 5 | 0 | 0 | 0 | 0 |
| 23 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 56 | 0 | 0 |
| 0 | 0 | 13 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 57 |
| 0 | 0 | 0 | 0 | 16 | 57 |
G:=sub<GL(6,GF(73))| [3,42,0,0,0,0,31,45,0,0,0,0,0,0,9,1,0,0,0,0,0,65,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,64,30,0,0,0,0,0,65,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[55,23,0,0,0,0,5,18,0,0,0,0,0,0,1,13,0,0,0,0,56,72,0,0,0,0,0,0,57,16,0,0,0,0,57,57] >;
C36.18D6 in GAP, Magma, Sage, TeX
C_{36}._{18}D_6 % in TeX
G:=Group("C36.18D6"); // GroupNames label
G:=SmallGroup(432,191);
// by ID
G=gap.SmallGroup(432,191);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,254,135,58,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^36=b^6=1,c^2=a^27,b*a*b^-1=a^19,c*a*c^-1=a^17,c*b*c^-1=a^27*b^-1>;
// generators/relations