metabelian, supersoluble, monomial
Aliases: C36.17D6, C12.17D18, D4.(C9⋊S3), (C3×D4).5D9, (D4×C9).5S3, (C3×C9)⋊12SD16, C9⋊3(D4.S3), C3⋊3(D4.D9), (C3×C18).47D4, (C3×C12).85D6, C12.D9⋊6C2, C36.S3⋊4C2, C6.24(C9⋊D4), (D4×C32).9S3, C18.24(C3⋊D4), (C3×C36).20C22, C3.(C32⋊9SD16), C6.16(C32⋊7D4), C32.5(D4.S3), C2.4(C6.D18), C4.1(C2×C9⋊S3), (D4×C3×C9).2C2, C12.1(C2×C3⋊S3), (C3×D4).1(C3⋊S3), (C3×C6).99(C3⋊D4), SmallGroup(432,190)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.17D6
G = < a,b,c | a36=b6=1, c2=a9, bab-1=a19, cac-1=a17, cbc-1=a27b-1 >
Subgroups: 480 in 100 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C8, D4, Q8, C9, C32, Dic3, C12, C12, C2×C6, SD16, C18, C18, C3×C6, C3×C6, C3⋊C8, Dic6, C3×D4, C3×D4, C3×C9, Dic9, C36, C2×C18, C3⋊Dic3, C3×C12, C62, D4.S3, C3×C18, C3×C18, C9⋊C8, Dic18, D4×C9, C32⋊4C8, C32⋊4Q8, D4×C32, C9⋊Dic3, C3×C36, C6×C18, D4.D9, C32⋊9SD16, C36.S3, C12.D9, D4×C3×C9, C36.17D6
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, C3⋊S3, C3⋊D4, D18, C2×C3⋊S3, D4.S3, C9⋊S3, C9⋊D4, C32⋊7D4, C2×C9⋊S3, D4.D9, C32⋊9SD16, C6.D18, C36.17D6
►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)
(1 156 56)(2 175 57 20 157 39)(3 158 58)(4 177 59 22 159 41)(5 160 60)(6 179 61 24 161 43)(7 162 62)(8 145 63 26 163 45)(9 164 64)(10 147 65 28 165 47)(11 166 66)(12 149 67 30 167 49)(13 168 68)(14 151 69 32 169 51)(15 170 70)(16 153 71 34 171 53)(17 172 72)(18 155 37 36 173 55)(19 174 38)(21 176 40)(23 178 42)(25 180 44)(27 146 46)(29 148 48)(31 150 50)(33 152 52)(35 154 54)(73 206 112 82 197 121)(74 189 113 101 198 140)(75 208 114 84 199 123)(76 191 115 103 200 142)(77 210 116 86 201 125)(78 193 117 105 202 144)(79 212 118 88 203 127)(80 195 119 107 204 110)(81 214 120 90 205 129)(83 216 122 92 207 131)(85 182 124 94 209 133)(87 184 126 96 211 135)(89 186 128 98 213 137)(91 188 130 100 215 139)(93 190 132 102 181 141)(95 192 134 104 183 143)(97 194 136 106 185 109)(99 196 138 108 187 111)
(1 126 10 135 19 144 28 117)(2 143 11 116 20 125 29 134)(3 124 12 133 21 142 30 115)(4 141 13 114 22 123 31 132)(5 122 14 131 23 140 32 113)(6 139 15 112 24 121 33 130)(7 120 16 129 25 138 34 111)(8 137 17 110 26 119 35 128)(9 118 18 127 27 136 36 109)(37 88 46 97 55 106 64 79)(38 105 47 78 56 87 65 96)(39 86 48 95 57 104 66 77)(40 103 49 76 58 85 67 94)(41 84 50 93 59 102 68 75)(42 101 51 74 60 83 69 92)(43 82 52 91 61 100 70 73)(44 99 53 108 62 81 71 90)(45 80 54 89 63 98 72 107)(145 204 154 213 163 186 172 195)(146 185 155 194 164 203 173 212)(147 202 156 211 165 184 174 193)(148 183 157 192 166 201 175 210)(149 200 158 209 167 182 176 191)(150 181 159 190 168 199 177 208)(151 198 160 207 169 216 178 189)(152 215 161 188 170 197 179 206)(153 196 162 205 171 214 180 187)
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), (1,156,56)(2,175,57,20,157,39)(3,158,58)(4,177,59,22,159,41)(5,160,60)(6,179,61,24,161,43)(7,162,62)(8,145,63,26,163,45)(9,164,64)(10,147,65,28,165,47)(11,166,66)(12,149,67,30,167,49)(13,168,68)(14,151,69,32,169,51)(15,170,70)(16,153,71,34,171,53)(17,172,72)(18,155,37,36,173,55)(19,174,38)(21,176,40)(23,178,42)(25,180,44)(27,146,46)(29,148,48)(31,150,50)(33,152,52)(35,154,54)(73,206,112,82,197,121)(74,189,113,101,198,140)(75,208,114,84,199,123)(76,191,115,103,200,142)(77,210,116,86,201,125)(78,193,117,105,202,144)(79,212,118,88,203,127)(80,195,119,107,204,110)(81,214,120,90,205,129)(83,216,122,92,207,131)(85,182,124,94,209,133)(87,184,126,96,211,135)(89,186,128,98,213,137)(91,188,130,100,215,139)(93,190,132,102,181,141)(95,192,134,104,183,143)(97,194,136,106,185,109)(99,196,138,108,187,111), (1,126,10,135,19,144,28,117)(2,143,11,116,20,125,29,134)(3,124,12,133,21,142,30,115)(4,141,13,114,22,123,31,132)(5,122,14,131,23,140,32,113)(6,139,15,112,24,121,33,130)(7,120,16,129,25,138,34,111)(8,137,17,110,26,119,35,128)(9,118,18,127,27,136,36,109)(37,88,46,97,55,106,64,79)(38,105,47,78,56,87,65,96)(39,86,48,95,57,104,66,77)(40,103,49,76,58,85,67,94)(41,84,50,93,59,102,68,75)(42,101,51,74,60,83,69,92)(43,82,52,91,61,100,70,73)(44,99,53,108,62,81,71,90)(45,80,54,89,63,98,72,107)(145,204,154,213,163,186,172,195)(146,185,155,194,164,203,173,212)(147,202,156,211,165,184,174,193)(148,183,157,192,166,201,175,210)(149,200,158,209,167,182,176,191)(150,181,159,190,168,199,177,208)(151,198,160,207,169,216,178,189)(152,215,161,188,170,197,179,206)(153,196,162,205,171,214,180,187)>;
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), (1,156,56)(2,175,57,20,157,39)(3,158,58)(4,177,59,22,159,41)(5,160,60)(6,179,61,24,161,43)(7,162,62)(8,145,63,26,163,45)(9,164,64)(10,147,65,28,165,47)(11,166,66)(12,149,67,30,167,49)(13,168,68)(14,151,69,32,169,51)(15,170,70)(16,153,71,34,171,53)(17,172,72)(18,155,37,36,173,55)(19,174,38)(21,176,40)(23,178,42)(25,180,44)(27,146,46)(29,148,48)(31,150,50)(33,152,52)(35,154,54)(73,206,112,82,197,121)(74,189,113,101,198,140)(75,208,114,84,199,123)(76,191,115,103,200,142)(77,210,116,86,201,125)(78,193,117,105,202,144)(79,212,118,88,203,127)(80,195,119,107,204,110)(81,214,120,90,205,129)(83,216,122,92,207,131)(85,182,124,94,209,133)(87,184,126,96,211,135)(89,186,128,98,213,137)(91,188,130,100,215,139)(93,190,132,102,181,141)(95,192,134,104,183,143)(97,194,136,106,185,109)(99,196,138,108,187,111), (1,126,10,135,19,144,28,117)(2,143,11,116,20,125,29,134)(3,124,12,133,21,142,30,115)(4,141,13,114,22,123,31,132)(5,122,14,131,23,140,32,113)(6,139,15,112,24,121,33,130)(7,120,16,129,25,138,34,111)(8,137,17,110,26,119,35,128)(9,118,18,127,27,136,36,109)(37,88,46,97,55,106,64,79)(38,105,47,78,56,87,65,96)(39,86,48,95,57,104,66,77)(40,103,49,76,58,85,67,94)(41,84,50,93,59,102,68,75)(42,101,51,74,60,83,69,92)(43,82,52,91,61,100,70,73)(44,99,53,108,62,81,71,90)(45,80,54,89,63,98,72,107)(145,204,154,213,163,186,172,195)(146,185,155,194,164,203,173,212)(147,202,156,211,165,184,174,193)(148,183,157,192,166,201,175,210)(149,200,158,209,167,182,176,191)(150,181,159,190,168,199,177,208)(151,198,160,207,169,216,178,189)(152,215,161,188,170,197,179,206)(153,196,162,205,171,214,180,187) );
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)], [(1,156,56),(2,175,57,20,157,39),(3,158,58),(4,177,59,22,159,41),(5,160,60),(6,179,61,24,161,43),(7,162,62),(8,145,63,26,163,45),(9,164,64),(10,147,65,28,165,47),(11,166,66),(12,149,67,30,167,49),(13,168,68),(14,151,69,32,169,51),(15,170,70),(16,153,71,34,171,53),(17,172,72),(18,155,37,36,173,55),(19,174,38),(21,176,40),(23,178,42),(25,180,44),(27,146,46),(29,148,48),(31,150,50),(33,152,52),(35,154,54),(73,206,112,82,197,121),(74,189,113,101,198,140),(75,208,114,84,199,123),(76,191,115,103,200,142),(77,210,116,86,201,125),(78,193,117,105,202,144),(79,212,118,88,203,127),(80,195,119,107,204,110),(81,214,120,90,205,129),(83,216,122,92,207,131),(85,182,124,94,209,133),(87,184,126,96,211,135),(89,186,128,98,213,137),(91,188,130,100,215,139),(93,190,132,102,181,141),(95,192,134,104,183,143),(97,194,136,106,185,109),(99,196,138,108,187,111)], [(1,126,10,135,19,144,28,117),(2,143,11,116,20,125,29,134),(3,124,12,133,21,142,30,115),(4,141,13,114,22,123,31,132),(5,122,14,131,23,140,32,113),(6,139,15,112,24,121,33,130),(7,120,16,129,25,138,34,111),(8,137,17,110,26,119,35,128),(9,118,18,127,27,136,36,109),(37,88,46,97,55,106,64,79),(38,105,47,78,56,87,65,96),(39,86,48,95,57,104,66,77),(40,103,49,76,58,85,67,94),(41,84,50,93,59,102,68,75),(42,101,51,74,60,83,69,92),(43,82,52,91,61,100,70,73),(44,99,53,108,62,81,71,90),(45,80,54,89,63,98,72,107),(145,204,154,213,163,186,172,195),(146,185,155,194,164,203,173,212),(147,202,156,211,165,184,174,193),(148,183,157,192,166,201,175,210),(149,200,158,209,167,182,176,191),(150,181,159,190,168,199,177,208),(151,198,160,207,169,216,178,189),(152,215,161,188,170,197,179,206),(153,196,162,205,171,214,180,187)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 8A | 8B | 9A | ··· | 9I | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18AA | 36A | ··· | 36I |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 108 | 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 | SD16 | D9 | C3⋊D4 | C3⋊D4 | D18 | C9⋊D4 | D4.S3 | D4.S3 | D4.D9 |
| kernel | C36.17D6 | C36.S3 | C12.D9 | 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 C36.17D6 ►in GL6(𝔽73)
| 69 | 60 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 42 |
| 0 | 0 | 0 | 0 | 66 | 72 |
| 64 | 16 | 0 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 66 | 72 |
| 20 | 15 | 0 | 0 | 0 | 0 |
| 56 | 53 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 68 | 0 | 0 |
| 0 | 0 | 29 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 33 |
| 0 | 0 | 0 | 0 | 31 | 61 |
G:=sub<GL(6,GF(73))| [69,0,0,0,0,0,60,18,0,0,0,0,0,0,16,0,0,0,0,0,0,32,0,0,0,0,0,0,1,66,0,0,0,0,42,72],[64,0,0,0,0,0,16,65,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,66,0,0,0,0,0,72],[20,56,0,0,0,0,15,53,0,0,0,0,0,0,0,29,0,0,0,0,68,0,0,0,0,0,0,0,0,31,0,0,0,0,33,61] >;
C36.17D6 in GAP, Magma, Sage, TeX
C_{36}._{17}D_6 % in TeX
G:=Group("C36.17D6"); // GroupNames label
G:=SmallGroup(432,190);
// by ID
G=gap.SmallGroup(432,190);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,254,135,58,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^36=b^6=1,c^2=a^9,b*a*b^-1=a^19,c*a*c^-1=a^17,c*b*c^-1=a^27*b^-1>;
// generators/relations