metabelian, supersoluble, monomial
Aliases: C36.69D6, C12.69D18, C36.3Dic3, C12.3Dic9, C62.19Dic3, (C3×C36).4C4, (C6×C18).7C4, C4.(C9⋊Dic3), (C3×C9)⋊9M4(2), (C6×C36).12C2, (C6×C12).35S3, (C2×C36).12S3, (C2×C12).12D9, (C2×C6).8Dic9, (C3×C12).219D6, C9⋊2(C4.Dic3), C3⋊2(C4.Dic9), (C2×C18).8Dic3, C6.13(C2×Dic9), C36.S3⋊12C2, C22.(C9⋊Dic3), (C3×C36).72C22, C12.1(C3⋊Dic3), C3.(C12.58D6), C18.13(C2×Dic3), (C3×C12).15Dic3, C32.4(C4.Dic3), C4.15(C2×C9⋊S3), C12.71(C2×C3⋊S3), (C2×C4).2(C9⋊S3), C6.7(C2×C3⋊Dic3), C2.3(C2×C9⋊Dic3), (C3×C18).37(C2×C4), (C2×C12).12(C3⋊S3), (C2×C6).8(C3⋊Dic3), (C3×C6).60(C2×Dic3), SmallGroup(432,179)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.69D6
G = < a,b,c | a36=b6=1, c2=a27, ab=ba, cac-1=a17, cbc-1=a18b-1 >
Subgroups: 308 in 100 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C9, C32, C12, C12, C2×C6, C2×C6, M4(2), C18, C18, C3×C6, C3×C6, C3⋊C8, C2×C12, C2×C12, C3×C9, C36, C2×C18, C3×C12, C62, C4.Dic3, C3×C18, C3×C18, C9⋊C8, C2×C36, C32⋊4C8, C6×C12, C3×C36, C6×C18, C4.Dic9, C12.58D6, C36.S3, C6×C36, C36.69D6
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, M4(2), D9, C3⋊S3, C2×Dic3, Dic9, D18, C3⋊Dic3, C2×C3⋊S3, C4.Dic3, C9⋊S3, C2×Dic9, C2×C3⋊Dic3, C9⋊Dic3, C2×C9⋊S3, C4.Dic9, C12.58D6, C2×C9⋊Dic3, C36.69D6
►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)
(1 191 67)(2 192 68)(3 193 69)(4 194 70)(5 195 71)(6 196 72)(7 197 37)(8 198 38)(9 199 39)(10 200 40)(11 201 41)(12 202 42)(13 203 43)(14 204 44)(15 205 45)(16 206 46)(17 207 47)(18 208 48)(19 209 49)(20 210 50)(21 211 51)(22 212 52)(23 213 53)(24 214 54)(25 215 55)(26 216 56)(27 181 57)(28 182 58)(29 183 59)(30 184 60)(31 185 61)(32 186 62)(33 187 63)(34 188 64)(35 189 65)(36 190 66)(73 180 142 91 162 124)(74 145 143 92 163 125)(75 146 144 93 164 126)(76 147 109 94 165 127)(77 148 110 95 166 128)(78 149 111 96 167 129)(79 150 112 97 168 130)(80 151 113 98 169 131)(81 152 114 99 170 132)(82 153 115 100 171 133)(83 154 116 101 172 134)(84 155 117 102 173 135)(85 156 118 103 174 136)(86 157 119 104 175 137)(87 158 120 105 176 138)(88 159 121 106 177 139)(89 160 122 107 178 140)(90 161 123 108 179 141)
(1 163 28 154 19 145 10 172)(2 180 29 171 20 162 11 153)(3 161 30 152 21 179 12 170)(4 178 31 169 22 160 13 151)(5 159 32 150 23 177 14 168)(6 176 33 167 24 158 15 149)(7 157 34 148 25 175 16 166)(8 174 35 165 26 156 17 147)(9 155 36 146 27 173 18 164)(37 137 64 128 55 119 46 110)(38 118 65 109 56 136 47 127)(39 135 66 126 57 117 48 144)(40 116 67 143 58 134 49 125)(41 133 68 124 59 115 50 142)(42 114 69 141 60 132 51 123)(43 131 70 122 61 113 52 140)(44 112 71 139 62 130 53 121)(45 129 72 120 63 111 54 138)(73 201 100 192 91 183 82 210)(74 182 101 209 92 200 83 191)(75 199 102 190 93 181 84 208)(76 216 103 207 94 198 85 189)(77 197 104 188 95 215 86 206)(78 214 105 205 96 196 87 187)(79 195 106 186 97 213 88 204)(80 212 107 203 98 194 89 185)(81 193 108 184 99 211 90 202)
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), (1,191,67)(2,192,68)(3,193,69)(4,194,70)(5,195,71)(6,196,72)(7,197,37)(8,198,38)(9,199,39)(10,200,40)(11,201,41)(12,202,42)(13,203,43)(14,204,44)(15,205,45)(16,206,46)(17,207,47)(18,208,48)(19,209,49)(20,210,50)(21,211,51)(22,212,52)(23,213,53)(24,214,54)(25,215,55)(26,216,56)(27,181,57)(28,182,58)(29,183,59)(30,184,60)(31,185,61)(32,186,62)(33,187,63)(34,188,64)(35,189,65)(36,190,66)(73,180,142,91,162,124)(74,145,143,92,163,125)(75,146,144,93,164,126)(76,147,109,94,165,127)(77,148,110,95,166,128)(78,149,111,96,167,129)(79,150,112,97,168,130)(80,151,113,98,169,131)(81,152,114,99,170,132)(82,153,115,100,171,133)(83,154,116,101,172,134)(84,155,117,102,173,135)(85,156,118,103,174,136)(86,157,119,104,175,137)(87,158,120,105,176,138)(88,159,121,106,177,139)(89,160,122,107,178,140)(90,161,123,108,179,141), (1,163,28,154,19,145,10,172)(2,180,29,171,20,162,11,153)(3,161,30,152,21,179,12,170)(4,178,31,169,22,160,13,151)(5,159,32,150,23,177,14,168)(6,176,33,167,24,158,15,149)(7,157,34,148,25,175,16,166)(8,174,35,165,26,156,17,147)(9,155,36,146,27,173,18,164)(37,137,64,128,55,119,46,110)(38,118,65,109,56,136,47,127)(39,135,66,126,57,117,48,144)(40,116,67,143,58,134,49,125)(41,133,68,124,59,115,50,142)(42,114,69,141,60,132,51,123)(43,131,70,122,61,113,52,140)(44,112,71,139,62,130,53,121)(45,129,72,120,63,111,54,138)(73,201,100,192,91,183,82,210)(74,182,101,209,92,200,83,191)(75,199,102,190,93,181,84,208)(76,216,103,207,94,198,85,189)(77,197,104,188,95,215,86,206)(78,214,105,205,96,196,87,187)(79,195,106,186,97,213,88,204)(80,212,107,203,98,194,89,185)(81,193,108,184,99,211,90,202)>;
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), (1,191,67)(2,192,68)(3,193,69)(4,194,70)(5,195,71)(6,196,72)(7,197,37)(8,198,38)(9,199,39)(10,200,40)(11,201,41)(12,202,42)(13,203,43)(14,204,44)(15,205,45)(16,206,46)(17,207,47)(18,208,48)(19,209,49)(20,210,50)(21,211,51)(22,212,52)(23,213,53)(24,214,54)(25,215,55)(26,216,56)(27,181,57)(28,182,58)(29,183,59)(30,184,60)(31,185,61)(32,186,62)(33,187,63)(34,188,64)(35,189,65)(36,190,66)(73,180,142,91,162,124)(74,145,143,92,163,125)(75,146,144,93,164,126)(76,147,109,94,165,127)(77,148,110,95,166,128)(78,149,111,96,167,129)(79,150,112,97,168,130)(80,151,113,98,169,131)(81,152,114,99,170,132)(82,153,115,100,171,133)(83,154,116,101,172,134)(84,155,117,102,173,135)(85,156,118,103,174,136)(86,157,119,104,175,137)(87,158,120,105,176,138)(88,159,121,106,177,139)(89,160,122,107,178,140)(90,161,123,108,179,141), (1,163,28,154,19,145,10,172)(2,180,29,171,20,162,11,153)(3,161,30,152,21,179,12,170)(4,178,31,169,22,160,13,151)(5,159,32,150,23,177,14,168)(6,176,33,167,24,158,15,149)(7,157,34,148,25,175,16,166)(8,174,35,165,26,156,17,147)(9,155,36,146,27,173,18,164)(37,137,64,128,55,119,46,110)(38,118,65,109,56,136,47,127)(39,135,66,126,57,117,48,144)(40,116,67,143,58,134,49,125)(41,133,68,124,59,115,50,142)(42,114,69,141,60,132,51,123)(43,131,70,122,61,113,52,140)(44,112,71,139,62,130,53,121)(45,129,72,120,63,111,54,138)(73,201,100,192,91,183,82,210)(74,182,101,209,92,200,83,191)(75,199,102,190,93,181,84,208)(76,216,103,207,94,198,85,189)(77,197,104,188,95,215,86,206)(78,214,105,205,96,196,87,187)(79,195,106,186,97,213,88,204)(80,212,107,203,98,194,89,185)(81,193,108,184,99,211,90,202) );
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)], [(1,191,67),(2,192,68),(3,193,69),(4,194,70),(5,195,71),(6,196,72),(7,197,37),(8,198,38),(9,199,39),(10,200,40),(11,201,41),(12,202,42),(13,203,43),(14,204,44),(15,205,45),(16,206,46),(17,207,47),(18,208,48),(19,209,49),(20,210,50),(21,211,51),(22,212,52),(23,213,53),(24,214,54),(25,215,55),(26,216,56),(27,181,57),(28,182,58),(29,183,59),(30,184,60),(31,185,61),(32,186,62),(33,187,63),(34,188,64),(35,189,65),(36,190,66),(73,180,142,91,162,124),(74,145,143,92,163,125),(75,146,144,93,164,126),(76,147,109,94,165,127),(77,148,110,95,166,128),(78,149,111,96,167,129),(79,150,112,97,168,130),(80,151,113,98,169,131),(81,152,114,99,170,132),(82,153,115,100,171,133),(83,154,116,101,172,134),(84,155,117,102,173,135),(85,156,118,103,174,136),(86,157,119,104,175,137),(87,158,120,105,176,138),(88,159,121,106,177,139),(89,160,122,107,178,140),(90,161,123,108,179,141)], [(1,163,28,154,19,145,10,172),(2,180,29,171,20,162,11,153),(3,161,30,152,21,179,12,170),(4,178,31,169,22,160,13,151),(5,159,32,150,23,177,14,168),(6,176,33,167,24,158,15,149),(7,157,34,148,25,175,16,166),(8,174,35,165,26,156,17,147),(9,155,36,146,27,173,18,164),(37,137,64,128,55,119,46,110),(38,118,65,109,56,136,47,127),(39,135,66,126,57,117,48,144),(40,116,67,143,58,134,49,125),(41,133,68,124,59,115,50,142),(42,114,69,141,60,132,51,123),(43,131,70,122,61,113,52,140),(44,112,71,139,62,130,53,121),(45,129,72,120,63,111,54,138),(73,201,100,192,91,183,82,210),(74,182,101,209,92,200,83,191),(75,199,102,190,93,181,84,208),(76,216,103,207,94,198,85,189),(77,197,104,188,95,215,86,206),(78,214,105,205,96,196,87,187),(79,195,106,186,97,213,88,204),(80,212,107,203,98,194,89,185),(81,193,108,184,99,211,90,202)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 9A | ··· | 9I | 12A | ··· | 12P | 18A | ··· | 18AA | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | ··· | 2 | 54 | 54 | 54 | 54 | 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 | C4 | C4 | S3 | S3 | Dic3 | D6 | Dic3 | Dic3 | D6 | Dic3 | M4(2) | D9 | Dic9 | D18 | Dic9 | C4.Dic3 | C4.Dic3 | C4.Dic9 |
| kernel | C36.69D6 | C36.S3 | C6×C36 | C3×C36 | C6×C18 | C2×C36 | C6×C12 | C36 | C36 | C2×C18 | C3×C12 | C3×C12 | C62 | C3×C9 | C2×C12 | C12 | C12 | C2×C6 | C9 | C32 | C3 |
| # reps | 1 | 2 | 1 | 2 | 2 | 3 | 1 | 3 | 3 | 3 | 1 | 1 | 1 | 2 | 9 | 9 | 9 | 9 | 12 | 4 | 36 |
Matrix representation of C36.69D6 ►in GL4(𝔽73) generated by
| 38 | 0 | 0 | 0 |
| 0 | 48 | 0 | 0 |
| 0 | 0 | 8 | 63 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 8 | 63 |
| 0 | 0 | 0 | 64 |
| 0 | 1 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 55 | 43 |
| 0 | 0 | 57 | 18 |
G:=sub<GL(4,GF(73))| [38,0,0,0,0,48,0,0,0,0,8,0,0,0,63,64],[1,0,0,0,0,72,0,0,0,0,8,0,0,0,63,64],[0,27,0,0,1,0,0,0,0,0,55,57,0,0,43,18] >;
C36.69D6 in GAP, Magma, Sage, TeX
C_{36}._{69}D_6 % in TeX
G:=Group("C36.69D6"); // GroupNames label
G:=SmallGroup(432,179);
// by ID
G=gap.SmallGroup(432,179);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,58,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^36=b^6=1,c^2=a^27,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=a^18*b^-1>;
// generators/relations