metabelian, supersoluble, monomial
Aliases: C36.70D6, C12.70D18, C62.137D6, (C2×C36)⋊4S3, (C6×C36)⋊9C2, (C2×C12)⋊4D9, C9⋊5(C4○D12), (C6×C12).36S3, (C2×C6).44D18, (C2×C18).44D6, C36⋊S3⋊11C2, (C3×C12).196D6, C3⋊5(D36⋊5C2), C12.D9⋊11C2, C6.42(C22×D9), C6.D18⋊7C2, (C3×C18).51C23, C18.42(C22×S3), (C6×C18).50C22, (C3×C36).64C22, C3.(C12.59D6), C32.7(C4○D12), C9⋊Dic3.14C22, (C4×C9⋊S3)⋊9C2, (C2×C4)⋊3(C9⋊S3), C4.16(C2×C9⋊S3), C12.73(C2×C3⋊S3), (C3×C9)⋊13(C4○D4), C22.2(C2×C9⋊S3), C2.5(C22×C9⋊S3), C6.31(C22×C3⋊S3), (C2×C12).13(C3⋊S3), (C2×C9⋊S3).12C22, (C3×C6).165(C22×S3), (C2×C6).37(C2×C3⋊S3), SmallGroup(432,383)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.70D6
G = < a,b,c | a36=1, b6=a24, c2=a18, ab=ba, cac-1=a17, cbc-1=a30b5 >
Subgroups: 1220 in 200 conjugacy classes, 71 normal (27 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, D9, C18, C18, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×C9, Dic9, C36, D18, C2×C18, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C4○D12, C9⋊S3, C3×C18, C3×C18, Dic18, C4×D9, D36, C9⋊D4, C2×C36, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C9⋊Dic3, C3×C36, C2×C9⋊S3, C6×C18, D36⋊5C2, C12.59D6, C12.D9, C4×C9⋊S3, C36⋊S3, C6.D18, C6×C36, C36.70D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C3⋊S3, C22×S3, D18, C2×C3⋊S3, C4○D12, C9⋊S3, C22×D9, C22×C3⋊S3, C2×C9⋊S3, D36⋊5C2, C12.59D6, C22×C9⋊S3, C36.70D6
►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)
(1 206 125 13 182 137 25 194 113)(2 207 126 14 183 138 26 195 114)(3 208 127 15 184 139 27 196 115)(4 209 128 16 185 140 28 197 116)(5 210 129 17 186 141 29 198 117)(6 211 130 18 187 142 30 199 118)(7 212 131 19 188 143 31 200 119)(8 213 132 20 189 144 32 201 120)(9 214 133 21 190 109 33 202 121)(10 215 134 22 191 110 34 203 122)(11 216 135 23 192 111 35 204 123)(12 181 136 24 193 112 36 205 124)(37 156 104 67 150 98 61 180 92 55 174 86 49 168 80 43 162 74)(38 157 105 68 151 99 62 145 93 56 175 87 50 169 81 44 163 75)(39 158 106 69 152 100 63 146 94 57 176 88 51 170 82 45 164 76)(40 159 107 70 153 101 64 147 95 58 177 89 52 171 83 46 165 77)(41 160 108 71 154 102 65 148 96 59 178 90 53 172 84 47 166 78)(42 161 73 72 155 103 66 149 97 60 179 91 54 173 85 48 167 79)
(1 149 19 167)(2 166 20 148)(3 147 21 165)(4 164 22 146)(5 145 23 163)(6 162 24 180)(7 179 25 161)(8 160 26 178)(9 177 27 159)(10 158 28 176)(11 175 29 157)(12 156 30 174)(13 173 31 155)(14 154 32 172)(15 171 33 153)(16 152 34 170)(17 169 35 151)(18 150 36 168)(37 181 55 199)(38 198 56 216)(39 215 57 197)(40 196 58 214)(41 213 59 195)(42 194 60 212)(43 211 61 193)(44 192 62 210)(45 209 63 191)(46 190 64 208)(47 207 65 189)(48 188 66 206)(49 205 67 187)(50 186 68 204)(51 203 69 185)(52 184 70 202)(53 201 71 183)(54 182 72 200)(73 137 91 119)(74 118 92 136)(75 135 93 117)(76 116 94 134)(77 133 95 115)(78 114 96 132)(79 131 97 113)(80 112 98 130)(81 129 99 111)(82 110 100 128)(83 127 101 109)(84 144 102 126)(85 125 103 143)(86 142 104 124)(87 123 105 141)(88 140 106 122)(89 121 107 139)(90 138 108 120)
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), (1,206,125,13,182,137,25,194,113)(2,207,126,14,183,138,26,195,114)(3,208,127,15,184,139,27,196,115)(4,209,128,16,185,140,28,197,116)(5,210,129,17,186,141,29,198,117)(6,211,130,18,187,142,30,199,118)(7,212,131,19,188,143,31,200,119)(8,213,132,20,189,144,32,201,120)(9,214,133,21,190,109,33,202,121)(10,215,134,22,191,110,34,203,122)(11,216,135,23,192,111,35,204,123)(12,181,136,24,193,112,36,205,124)(37,156,104,67,150,98,61,180,92,55,174,86,49,168,80,43,162,74)(38,157,105,68,151,99,62,145,93,56,175,87,50,169,81,44,163,75)(39,158,106,69,152,100,63,146,94,57,176,88,51,170,82,45,164,76)(40,159,107,70,153,101,64,147,95,58,177,89,52,171,83,46,165,77)(41,160,108,71,154,102,65,148,96,59,178,90,53,172,84,47,166,78)(42,161,73,72,155,103,66,149,97,60,179,91,54,173,85,48,167,79), (1,149,19,167)(2,166,20,148)(3,147,21,165)(4,164,22,146)(5,145,23,163)(6,162,24,180)(7,179,25,161)(8,160,26,178)(9,177,27,159)(10,158,28,176)(11,175,29,157)(12,156,30,174)(13,173,31,155)(14,154,32,172)(15,171,33,153)(16,152,34,170)(17,169,35,151)(18,150,36,168)(37,181,55,199)(38,198,56,216)(39,215,57,197)(40,196,58,214)(41,213,59,195)(42,194,60,212)(43,211,61,193)(44,192,62,210)(45,209,63,191)(46,190,64,208)(47,207,65,189)(48,188,66,206)(49,205,67,187)(50,186,68,204)(51,203,69,185)(52,184,70,202)(53,201,71,183)(54,182,72,200)(73,137,91,119)(74,118,92,136)(75,135,93,117)(76,116,94,134)(77,133,95,115)(78,114,96,132)(79,131,97,113)(80,112,98,130)(81,129,99,111)(82,110,100,128)(83,127,101,109)(84,144,102,126)(85,125,103,143)(86,142,104,124)(87,123,105,141)(88,140,106,122)(89,121,107,139)(90,138,108,120)>;
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), (1,206,125,13,182,137,25,194,113)(2,207,126,14,183,138,26,195,114)(3,208,127,15,184,139,27,196,115)(4,209,128,16,185,140,28,197,116)(5,210,129,17,186,141,29,198,117)(6,211,130,18,187,142,30,199,118)(7,212,131,19,188,143,31,200,119)(8,213,132,20,189,144,32,201,120)(9,214,133,21,190,109,33,202,121)(10,215,134,22,191,110,34,203,122)(11,216,135,23,192,111,35,204,123)(12,181,136,24,193,112,36,205,124)(37,156,104,67,150,98,61,180,92,55,174,86,49,168,80,43,162,74)(38,157,105,68,151,99,62,145,93,56,175,87,50,169,81,44,163,75)(39,158,106,69,152,100,63,146,94,57,176,88,51,170,82,45,164,76)(40,159,107,70,153,101,64,147,95,58,177,89,52,171,83,46,165,77)(41,160,108,71,154,102,65,148,96,59,178,90,53,172,84,47,166,78)(42,161,73,72,155,103,66,149,97,60,179,91,54,173,85,48,167,79), (1,149,19,167)(2,166,20,148)(3,147,21,165)(4,164,22,146)(5,145,23,163)(6,162,24,180)(7,179,25,161)(8,160,26,178)(9,177,27,159)(10,158,28,176)(11,175,29,157)(12,156,30,174)(13,173,31,155)(14,154,32,172)(15,171,33,153)(16,152,34,170)(17,169,35,151)(18,150,36,168)(37,181,55,199)(38,198,56,216)(39,215,57,197)(40,196,58,214)(41,213,59,195)(42,194,60,212)(43,211,61,193)(44,192,62,210)(45,209,63,191)(46,190,64,208)(47,207,65,189)(48,188,66,206)(49,205,67,187)(50,186,68,204)(51,203,69,185)(52,184,70,202)(53,201,71,183)(54,182,72,200)(73,137,91,119)(74,118,92,136)(75,135,93,117)(76,116,94,134)(77,133,95,115)(78,114,96,132)(79,131,97,113)(80,112,98,130)(81,129,99,111)(82,110,100,128)(83,127,101,109)(84,144,102,126)(85,125,103,143)(86,142,104,124)(87,123,105,141)(88,140,106,122)(89,121,107,139)(90,138,108,120) );
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)], [(1,206,125,13,182,137,25,194,113),(2,207,126,14,183,138,26,195,114),(3,208,127,15,184,139,27,196,115),(4,209,128,16,185,140,28,197,116),(5,210,129,17,186,141,29,198,117),(6,211,130,18,187,142,30,199,118),(7,212,131,19,188,143,31,200,119),(8,213,132,20,189,144,32,201,120),(9,214,133,21,190,109,33,202,121),(10,215,134,22,191,110,34,203,122),(11,216,135,23,192,111,35,204,123),(12,181,136,24,193,112,36,205,124),(37,156,104,67,150,98,61,180,92,55,174,86,49,168,80,43,162,74),(38,157,105,68,151,99,62,145,93,56,175,87,50,169,81,44,163,75),(39,158,106,69,152,100,63,146,94,57,176,88,51,170,82,45,164,76),(40,159,107,70,153,101,64,147,95,58,177,89,52,171,83,46,165,77),(41,160,108,71,154,102,65,148,96,59,178,90,53,172,84,47,166,78),(42,161,73,72,155,103,66,149,97,60,179,91,54,173,85,48,167,79)], [(1,149,19,167),(2,166,20,148),(3,147,21,165),(4,164,22,146),(5,145,23,163),(6,162,24,180),(7,179,25,161),(8,160,26,178),(9,177,27,159),(10,158,28,176),(11,175,29,157),(12,156,30,174),(13,173,31,155),(14,154,32,172),(15,171,33,153),(16,152,34,170),(17,169,35,151),(18,150,36,168),(37,181,55,199),(38,198,56,216),(39,215,57,197),(40,196,58,214),(41,213,59,195),(42,194,60,212),(43,211,61,193),(44,192,62,210),(45,209,63,191),(46,190,64,208),(47,207,65,189),(48,188,66,206),(49,205,67,187),(50,186,68,204),(51,203,69,185),(52,184,70,202),(53,201,71,183),(54,182,72,200),(73,137,91,119),(74,118,92,136),(75,135,93,117),(76,116,94,134),(77,133,95,115),(78,114,96,132),(79,131,97,113),(80,112,98,130),(81,129,99,111),(82,110,100,128),(83,127,101,109),(84,144,102,126),(85,125,103,143),(86,142,104,124),(87,123,105,141),(88,140,106,122),(89,121,107,139),(90,138,108,120)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6L | 9A | ··· | 9I | 12A | ··· | 12P | 18A | ··· | 18AA | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 54 | 54 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 54 | 54 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | C4○D4 | D9 | D18 | D18 | C4○D12 | C4○D12 | D36⋊5C2 |
| kernel | C36.70D6 | C12.D9 | C4×C9⋊S3 | C36⋊S3 | C6.D18 | C6×C36 | C2×C36 | C6×C12 | C36 | C2×C18 | C3×C12 | C62 | C3×C9 | C2×C12 | C12 | C2×C6 | C9 | C32 | C3 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 1 | 6 | 3 | 2 | 1 | 2 | 9 | 18 | 9 | 12 | 4 | 36 |
Matrix representation of C36.70D6 ►in GL4(𝔽37) generated by
| 14 | 0 | 0 | 0 |
| 0 | 29 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 2 | 34 |
| 26 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 2 | 34 |
| 0 | 10 | 0 | 0 |
| 11 | 0 | 0 | 0 |
| 0 | 0 | 30 | 34 |
| 0 | 0 | 16 | 7 |
G:=sub<GL(4,GF(37))| [14,0,0,0,0,29,0,0,0,0,12,2,0,0,0,34],[26,0,0,0,0,27,0,0,0,0,12,2,0,0,0,34],[0,11,0,0,10,0,0,0,0,0,30,16,0,0,34,7] >;
C36.70D6 in GAP, Magma, Sage, TeX
C_{36}._{70}D_6 % in TeX
G:=Group("C36.70D6"); // GroupNames label
G:=SmallGroup(432,383);
// by ID
G=gap.SmallGroup(432,383);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^36=1,b^6=a^24,c^2=a^18,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=a^30*b^5>;
// generators/relations