metabelian, supersoluble, monomial
Aliases: C36.29D6, C12.29D18, (Q8×C9)⋊5S3, (C3×Q8)⋊5D9, Q8⋊4(C9⋊S3), C36⋊S3⋊10C2, C9⋊3(Q8⋊3S3), C3⋊3(Q8⋊3D9), (C3×C12).108D6, C6.46(C22×D9), (C3×C36).35C22, (C3×C18).55C23, C18.46(C22×S3), C3.(C12.26D6), (Q8×C32).25S3, C9⋊Dic3.17C22, C32.5(Q8⋊3S3), (C4×C9⋊S3)⋊6C2, (Q8×C3×C9)⋊6C2, C4.7(C2×C9⋊S3), C12.8(C2×C3⋊S3), (C3×C9)⋊19(C4○D4), C2.9(C22×C9⋊S3), C6.35(C22×C3⋊S3), (C2×C9⋊S3).14C22, (C3×Q8).12(C3⋊S3), (C3×C6).169(C22×S3), SmallGroup(432,393)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.29D6
G = < a,b,c | a36=c2=1, b6=a6, bab-1=a19, cac=a-1, cbc=a30b5 >
Subgroups: 1352 in 200 conjugacy classes, 71 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C32, Dic3, C12, D6, C4○D4, D9, C18, C3⋊S3, C3×C6, C4×S3, D12, C3×Q8, C3×Q8, C3×C9, Dic9, C36, D18, C3⋊Dic3, C3×C12, C2×C3⋊S3, Q8⋊3S3, C9⋊S3, C3×C18, C4×D9, D36, Q8×C9, C4×C3⋊S3, C12⋊S3, Q8×C32, C9⋊Dic3, C3×C36, C2×C9⋊S3, Q8⋊3D9, C12.26D6, C4×C9⋊S3, C36⋊S3, Q8×C3×C9, C36.29D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C3⋊S3, C22×S3, D18, C2×C3⋊S3, Q8⋊3S3, C9⋊S3, C22×D9, C22×C3⋊S3, C2×C9⋊S3, Q8⋊3D9, C12.26D6, C22×C9⋊S3, C36.29D6
►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 70 117 211 172 84 7 40 123 181 178 90 13 46 129 187 148 96 19 52 135 193 154 102 25 58 141 199 160 108 31 64 111 205 166 78)(2 53 118 194 173 103 8 59 124 200 179 73 14 65 130 206 149 79 20 71 136 212 155 85 26 41 142 182 161 91 32 47 112 188 167 97)(3 72 119 213 174 86 9 42 125 183 180 92 15 48 131 189 150 98 21 54 137 195 156 104 27 60 143 201 162 74 33 66 113 207 168 80)(4 55 120 196 175 105 10 61 126 202 145 75 16 67 132 208 151 81 22 37 138 214 157 87 28 43 144 184 163 93 34 49 114 190 169 99)(5 38 121 215 176 88 11 44 127 185 146 94 17 50 133 191 152 100 23 56 139 197 158 106 29 62 109 203 164 76 35 68 115 209 170 82)(6 57 122 198 177 107 12 63 128 204 147 77 18 69 134 210 153 83 24 39 140 216 159 89 30 45 110 186 165 95 36 51 116 192 171 101)
(1 135)(2 134)(3 133)(4 132)(5 131)(6 130)(7 129)(8 128)(9 127)(10 126)(11 125)(12 124)(13 123)(14 122)(15 121)(16 120)(17 119)(18 118)(19 117)(20 116)(21 115)(22 114)(23 113)(24 112)(25 111)(26 110)(27 109)(28 144)(29 143)(30 142)(31 141)(32 140)(33 139)(34 138)(35 137)(36 136)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(73 198)(74 197)(75 196)(76 195)(77 194)(78 193)(79 192)(80 191)(81 190)(82 189)(83 188)(84 187)(85 186)(86 185)(87 184)(88 183)(89 182)(90 181)(91 216)(92 215)(93 214)(94 213)(95 212)(96 211)(97 210)(98 209)(99 208)(100 207)(101 206)(102 205)(103 204)(104 203)(105 202)(106 201)(107 200)(108 199)(145 175)(146 174)(147 173)(148 172)(149 171)(150 170)(151 169)(152 168)(153 167)(154 166)(155 165)(156 164)(157 163)(158 162)(159 161)(176 180)(177 179)
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), (1,70,117,211,172,84,7,40,123,181,178,90,13,46,129,187,148,96,19,52,135,193,154,102,25,58,141,199,160,108,31,64,111,205,166,78)(2,53,118,194,173,103,8,59,124,200,179,73,14,65,130,206,149,79,20,71,136,212,155,85,26,41,142,182,161,91,32,47,112,188,167,97)(3,72,119,213,174,86,9,42,125,183,180,92,15,48,131,189,150,98,21,54,137,195,156,104,27,60,143,201,162,74,33,66,113,207,168,80)(4,55,120,196,175,105,10,61,126,202,145,75,16,67,132,208,151,81,22,37,138,214,157,87,28,43,144,184,163,93,34,49,114,190,169,99)(5,38,121,215,176,88,11,44,127,185,146,94,17,50,133,191,152,100,23,56,139,197,158,106,29,62,109,203,164,76,35,68,115,209,170,82)(6,57,122,198,177,107,12,63,128,204,147,77,18,69,134,210,153,83,24,39,140,216,159,89,30,45,110,186,165,95,36,51,116,192,171,101), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,120)(17,119)(18,118)(19,117)(20,116)(21,115)(22,114)(23,113)(24,112)(25,111)(26,110)(27,109)(28,144)(29,143)(30,142)(31,141)(32,140)(33,139)(34,138)(35,137)(36,136)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(73,198)(74,197)(75,196)(76,195)(77,194)(78,193)(79,192)(80,191)(81,190)(82,189)(83,188)(84,187)(85,186)(86,185)(87,184)(88,183)(89,182)(90,181)(91,216)(92,215)(93,214)(94,213)(95,212)(96,211)(97,210)(98,209)(99,208)(100,207)(101,206)(102,205)(103,204)(104,203)(105,202)(106,201)(107,200)(108,199)(145,175)(146,174)(147,173)(148,172)(149,171)(150,170)(151,169)(152,168)(153,167)(154,166)(155,165)(156,164)(157,163)(158,162)(159,161)(176,180)(177,179)>;
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), (1,70,117,211,172,84,7,40,123,181,178,90,13,46,129,187,148,96,19,52,135,193,154,102,25,58,141,199,160,108,31,64,111,205,166,78)(2,53,118,194,173,103,8,59,124,200,179,73,14,65,130,206,149,79,20,71,136,212,155,85,26,41,142,182,161,91,32,47,112,188,167,97)(3,72,119,213,174,86,9,42,125,183,180,92,15,48,131,189,150,98,21,54,137,195,156,104,27,60,143,201,162,74,33,66,113,207,168,80)(4,55,120,196,175,105,10,61,126,202,145,75,16,67,132,208,151,81,22,37,138,214,157,87,28,43,144,184,163,93,34,49,114,190,169,99)(5,38,121,215,176,88,11,44,127,185,146,94,17,50,133,191,152,100,23,56,139,197,158,106,29,62,109,203,164,76,35,68,115,209,170,82)(6,57,122,198,177,107,12,63,128,204,147,77,18,69,134,210,153,83,24,39,140,216,159,89,30,45,110,186,165,95,36,51,116,192,171,101), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,120)(17,119)(18,118)(19,117)(20,116)(21,115)(22,114)(23,113)(24,112)(25,111)(26,110)(27,109)(28,144)(29,143)(30,142)(31,141)(32,140)(33,139)(34,138)(35,137)(36,136)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(73,198)(74,197)(75,196)(76,195)(77,194)(78,193)(79,192)(80,191)(81,190)(82,189)(83,188)(84,187)(85,186)(86,185)(87,184)(88,183)(89,182)(90,181)(91,216)(92,215)(93,214)(94,213)(95,212)(96,211)(97,210)(98,209)(99,208)(100,207)(101,206)(102,205)(103,204)(104,203)(105,202)(106,201)(107,200)(108,199)(145,175)(146,174)(147,173)(148,172)(149,171)(150,170)(151,169)(152,168)(153,167)(154,166)(155,165)(156,164)(157,163)(158,162)(159,161)(176,180)(177,179) );
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)], [(1,70,117,211,172,84,7,40,123,181,178,90,13,46,129,187,148,96,19,52,135,193,154,102,25,58,141,199,160,108,31,64,111,205,166,78),(2,53,118,194,173,103,8,59,124,200,179,73,14,65,130,206,149,79,20,71,136,212,155,85,26,41,142,182,161,91,32,47,112,188,167,97),(3,72,119,213,174,86,9,42,125,183,180,92,15,48,131,189,150,98,21,54,137,195,156,104,27,60,143,201,162,74,33,66,113,207,168,80),(4,55,120,196,175,105,10,61,126,202,145,75,16,67,132,208,151,81,22,37,138,214,157,87,28,43,144,184,163,93,34,49,114,190,169,99),(5,38,121,215,176,88,11,44,127,185,146,94,17,50,133,191,152,100,23,56,139,197,158,106,29,62,109,203,164,76,35,68,115,209,170,82),(6,57,122,198,177,107,12,63,128,204,147,77,18,69,134,210,153,83,24,39,140,216,159,89,30,45,110,186,165,95,36,51,116,192,171,101)], [(1,135),(2,134),(3,133),(4,132),(5,131),(6,130),(7,129),(8,128),(9,127),(10,126),(11,125),(12,124),(13,123),(14,122),(15,121),(16,120),(17,119),(18,118),(19,117),(20,116),(21,115),(22,114),(23,113),(24,112),(25,111),(26,110),(27,109),(28,144),(29,143),(30,142),(31,141),(32,140),(33,139),(34,138),(35,137),(36,136),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(50,72),(51,71),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(73,198),(74,197),(75,196),(76,195),(77,194),(78,193),(79,192),(80,191),(81,190),(82,189),(83,188),(84,187),(85,186),(86,185),(87,184),(88,183),(89,182),(90,181),(91,216),(92,215),(93,214),(94,213),(95,212),(96,211),(97,210),(98,209),(99,208),(100,207),(101,206),(102,205),(103,204),(104,203),(105,202),(106,201),(107,200),(108,199),(145,175),(146,174),(147,173),(148,172),(149,171),(150,170),(151,169),(152,168),(153,167),(154,166),(155,165),(156,164),(157,163),(158,162),(159,161),(176,180),(177,179)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 9A | ··· | 9I | 12A | ··· | 12L | 18A | ··· | 18I | 36A | ··· | 36AA |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 54 | 54 | 54 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 27 | 27 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | C4○D4 | D9 | D18 | Q8⋊3S3 | Q8⋊3S3 | Q8⋊3D9 |
| kernel | C36.29D6 | C4×C9⋊S3 | C36⋊S3 | Q8×C3×C9 | Q8×C9 | Q8×C32 | C36 | C3×C12 | C3×C9 | C3×Q8 | C12 | C9 | C32 | C3 |
| # reps | 1 | 3 | 3 | 1 | 3 | 1 | 9 | 3 | 2 | 9 | 27 | 3 | 1 | 9 |
Matrix representation of C36.29D6 ►in GL6(𝔽37)
| 20 | 6 | 0 | 0 | 0 | 0 |
| 31 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 31 |
| 0 | 0 | 0 | 0 | 0 | 31 |
| 11 | 20 | 0 | 0 | 0 | 0 |
| 17 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 1 |
| 0 | 0 | 0 | 0 | 35 | 1 |
| 31 | 17 | 0 | 0 | 0 | 0 |
| 11 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 2 | 36 |
G:=sub<GL(6,GF(37))| [20,31,0,0,0,0,6,26,0,0,0,0,0,0,0,1,0,0,0,0,36,1,0,0,0,0,0,0,6,0,0,0,0,0,31,31],[11,17,0,0,0,0,20,31,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,35,0,0,0,0,1,1],[31,11,0,0,0,0,17,6,0,0,0,0,0,0,1,36,0,0,0,0,0,36,0,0,0,0,0,0,1,2,0,0,0,0,0,36] >;
C36.29D6 in GAP, Magma, Sage, TeX
C_{36}._{29}D_6 % in TeX
G:=Group("C36.29D6"); // GroupNames label
G:=SmallGroup(432,393);
// by ID
G=gap.SmallGroup(432,393);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,58,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^36=c^2=1,b^6=a^6,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=a^30*b^5>;
// generators/relations