Copied to
clipboard

## G = C36.20D6order 432 = 24·33

### 20th non-split extension by C36 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C36 — C36.20D6
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C3×C36 — C36⋊S3 — C36.20D6
 Lower central C3×C9 — C3×C18 — C3×C36 — C36.20D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for C36.20D6
G = < a,b,c | a36=1, b6=a18, c2=a9, bab-1=a19, cac-1=a17, cbc-1=a27b5 >

Subgroups: 744 in 100 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C32, C12, C12, C12, D6, SD16, D9, C18, C3⋊S3, C3×C6, C3⋊C8, D12, C3×Q8, C3×Q8, C3×C9, C36, C36, D18, C3×C12, C3×C12, C2×C3⋊S3, Q82S3, C9⋊S3, C3×C18, C9⋊C8, D36, Q8×C9, C324C8, C12⋊S3, Q8×C32, C3×C36, C3×C36, C2×C9⋊S3, Q82D9, C3211SD16, C36.S3, C36⋊S3, Q8×C3×C9, C36.20D6
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, C3⋊S3, C3⋊D4, D18, C2×C3⋊S3, Q82S3, C9⋊S3, C9⋊D4, C327D4, C2×C9⋊S3, Q82D9, C3211SD16, C6.D18, C36.20D6

Smallest permutation representation of C36.20D6
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)
(1 182 113 179 56 100 19 200 131 161 38 82)(2 201 114 162 57 83 20 183 132 180 39 101)(3 184 115 145 58 102 21 202 133 163 40 84)(4 203 116 164 59 85 22 185 134 146 41 103)(5 186 117 147 60 104 23 204 135 165 42 86)(6 205 118 166 61 87 24 187 136 148 43 105)(7 188 119 149 62 106 25 206 137 167 44 88)(8 207 120 168 63 89 26 189 138 150 45 107)(9 190 121 151 64 108 27 208 139 169 46 90)(10 209 122 170 65 91 28 191 140 152 47 73)(11 192 123 153 66 74 29 210 141 171 48 92)(12 211 124 172 67 93 30 193 142 154 49 75)(13 194 125 155 68 76 31 212 143 173 50 94)(14 213 126 174 69 95 32 195 144 156 51 77)(15 196 127 157 70 78 33 214 109 175 52 96)(16 215 128 176 71 97 34 197 110 158 53 79)(17 198 129 159 72 80 35 216 111 177 54 98)(18 181 130 178 37 99 36 199 112 160 55 81)
(1 100 10 73 19 82 28 91)(2 81 11 90 20 99 29 108)(3 98 12 107 21 80 30 89)(4 79 13 88 22 97 31 106)(5 96 14 105 23 78 32 87)(6 77 15 86 24 95 33 104)(7 94 16 103 25 76 34 85)(8 75 17 84 26 93 35 102)(9 92 18 101 27 74 36 83)(37 183 46 192 55 201 64 210)(38 200 47 209 56 182 65 191)(39 181 48 190 57 199 66 208)(40 198 49 207 58 216 67 189)(41 215 50 188 59 197 68 206)(42 196 51 205 60 214 69 187)(43 213 52 186 61 195 70 204)(44 194 53 203 62 212 71 185)(45 211 54 184 63 193 72 202)(109 147 118 156 127 165 136 174)(110 164 119 173 128 146 137 155)(111 145 120 154 129 163 138 172)(112 162 121 171 130 180 139 153)(113 179 122 152 131 161 140 170)(114 160 123 169 132 178 141 151)(115 177 124 150 133 159 142 168)(116 158 125 167 134 176 143 149)(117 175 126 148 135 157 144 166)```

`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), (1,182,113,179,56,100,19,200,131,161,38,82)(2,201,114,162,57,83,20,183,132,180,39,101)(3,184,115,145,58,102,21,202,133,163,40,84)(4,203,116,164,59,85,22,185,134,146,41,103)(5,186,117,147,60,104,23,204,135,165,42,86)(6,205,118,166,61,87,24,187,136,148,43,105)(7,188,119,149,62,106,25,206,137,167,44,88)(8,207,120,168,63,89,26,189,138,150,45,107)(9,190,121,151,64,108,27,208,139,169,46,90)(10,209,122,170,65,91,28,191,140,152,47,73)(11,192,123,153,66,74,29,210,141,171,48,92)(12,211,124,172,67,93,30,193,142,154,49,75)(13,194,125,155,68,76,31,212,143,173,50,94)(14,213,126,174,69,95,32,195,144,156,51,77)(15,196,127,157,70,78,33,214,109,175,52,96)(16,215,128,176,71,97,34,197,110,158,53,79)(17,198,129,159,72,80,35,216,111,177,54,98)(18,181,130,178,37,99,36,199,112,160,55,81), (1,100,10,73,19,82,28,91)(2,81,11,90,20,99,29,108)(3,98,12,107,21,80,30,89)(4,79,13,88,22,97,31,106)(5,96,14,105,23,78,32,87)(6,77,15,86,24,95,33,104)(7,94,16,103,25,76,34,85)(8,75,17,84,26,93,35,102)(9,92,18,101,27,74,36,83)(37,183,46,192,55,201,64,210)(38,200,47,209,56,182,65,191)(39,181,48,190,57,199,66,208)(40,198,49,207,58,216,67,189)(41,215,50,188,59,197,68,206)(42,196,51,205,60,214,69,187)(43,213,52,186,61,195,70,204)(44,194,53,203,62,212,71,185)(45,211,54,184,63,193,72,202)(109,147,118,156,127,165,136,174)(110,164,119,173,128,146,137,155)(111,145,120,154,129,163,138,172)(112,162,121,171,130,180,139,153)(113,179,122,152,131,161,140,170)(114,160,123,169,132,178,141,151)(115,177,124,150,133,159,142,168)(116,158,125,167,134,176,143,149)(117,175,126,148,135,157,144,166)>;`

`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), (1,182,113,179,56,100,19,200,131,161,38,82)(2,201,114,162,57,83,20,183,132,180,39,101)(3,184,115,145,58,102,21,202,133,163,40,84)(4,203,116,164,59,85,22,185,134,146,41,103)(5,186,117,147,60,104,23,204,135,165,42,86)(6,205,118,166,61,87,24,187,136,148,43,105)(7,188,119,149,62,106,25,206,137,167,44,88)(8,207,120,168,63,89,26,189,138,150,45,107)(9,190,121,151,64,108,27,208,139,169,46,90)(10,209,122,170,65,91,28,191,140,152,47,73)(11,192,123,153,66,74,29,210,141,171,48,92)(12,211,124,172,67,93,30,193,142,154,49,75)(13,194,125,155,68,76,31,212,143,173,50,94)(14,213,126,174,69,95,32,195,144,156,51,77)(15,196,127,157,70,78,33,214,109,175,52,96)(16,215,128,176,71,97,34,197,110,158,53,79)(17,198,129,159,72,80,35,216,111,177,54,98)(18,181,130,178,37,99,36,199,112,160,55,81), (1,100,10,73,19,82,28,91)(2,81,11,90,20,99,29,108)(3,98,12,107,21,80,30,89)(4,79,13,88,22,97,31,106)(5,96,14,105,23,78,32,87)(6,77,15,86,24,95,33,104)(7,94,16,103,25,76,34,85)(8,75,17,84,26,93,35,102)(9,92,18,101,27,74,36,83)(37,183,46,192,55,201,64,210)(38,200,47,209,56,182,65,191)(39,181,48,190,57,199,66,208)(40,198,49,207,58,216,67,189)(41,215,50,188,59,197,68,206)(42,196,51,205,60,214,69,187)(43,213,52,186,61,195,70,204)(44,194,53,203,62,212,71,185)(45,211,54,184,63,193,72,202)(109,147,118,156,127,165,136,174)(110,164,119,173,128,146,137,155)(111,145,120,154,129,163,138,172)(112,162,121,171,130,180,139,153)(113,179,122,152,131,161,140,170)(114,160,123,169,132,178,141,151)(115,177,124,150,133,159,142,168)(116,158,125,167,134,176,143,149)(117,175,126,148,135,157,144,166) );`

`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)], [(1,182,113,179,56,100,19,200,131,161,38,82),(2,201,114,162,57,83,20,183,132,180,39,101),(3,184,115,145,58,102,21,202,133,163,40,84),(4,203,116,164,59,85,22,185,134,146,41,103),(5,186,117,147,60,104,23,204,135,165,42,86),(6,205,118,166,61,87,24,187,136,148,43,105),(7,188,119,149,62,106,25,206,137,167,44,88),(8,207,120,168,63,89,26,189,138,150,45,107),(9,190,121,151,64,108,27,208,139,169,46,90),(10,209,122,170,65,91,28,191,140,152,47,73),(11,192,123,153,66,74,29,210,141,171,48,92),(12,211,124,172,67,93,30,193,142,154,49,75),(13,194,125,155,68,76,31,212,143,173,50,94),(14,213,126,174,69,95,32,195,144,156,51,77),(15,196,127,157,70,78,33,214,109,175,52,96),(16,215,128,176,71,97,34,197,110,158,53,79),(17,198,129,159,72,80,35,216,111,177,54,98),(18,181,130,178,37,99,36,199,112,160,55,81)], [(1,100,10,73,19,82,28,91),(2,81,11,90,20,99,29,108),(3,98,12,107,21,80,30,89),(4,79,13,88,22,97,31,106),(5,96,14,105,23,78,32,87),(6,77,15,86,24,95,33,104),(7,94,16,103,25,76,34,85),(8,75,17,84,26,93,35,102),(9,92,18,101,27,74,36,83),(37,183,46,192,55,201,64,210),(38,200,47,209,56,182,65,191),(39,181,48,190,57,199,66,208),(40,198,49,207,58,216,67,189),(41,215,50,188,59,197,68,206),(42,196,51,205,60,214,69,187),(43,213,52,186,61,195,70,204),(44,194,53,203,62,212,71,185),(45,211,54,184,63,193,72,202),(109,147,118,156,127,165,136,174),(110,164,119,173,128,146,137,155),(111,145,120,154,129,163,138,172),(112,162,121,171,130,180,139,153),(113,179,122,152,131,161,140,170),(114,160,123,169,132,178,141,151),(115,177,124,150,133,159,142,168),(116,158,125,167,134,176,143,149),(117,175,126,148,135,157,144,166)]])`

72 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 8A 8B 9A ··· 9I 12A ··· 12L 18A ··· 18I 36A ··· 36AA order 1 2 2 3 3 3 3 4 4 6 6 6 6 8 8 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 108 2 2 2 2 2 4 2 2 2 2 54 54 2 ··· 2 4 ··· 4 2 ··· 2 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 Q8⋊2S3 Q8⋊2S3 Q8⋊2D9 kernel C36.20D6 C36.S3 C36⋊S3 Q8×C3×C9 Q8×C9 Q8×C32 C3×C18 C36 C3×C12 C3×C9 C3×Q8 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 C36.20D6 in GL6(𝔽73)

 72 72 0 0 0 0 1 0 0 0 0 0 0 0 31 70 0 0 0 0 3 28 0 0 0 0 0 0 67 3 0 0 0 0 12 6
,
 1 1 0 0 0 0 72 0 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 30 55 0 0 0 0 46 43
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 28 3 0 0 0 0 31 45 0 0 0 0 0 0 31 18 0 0 0 0 72 30

`G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,31,3,0,0,0,0,70,28,0,0,0,0,0,0,67,12,0,0,0,0,3,6],[1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,30,46,0,0,0,0,55,43],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,28,31,0,0,0,0,3,45,0,0,0,0,0,0,31,72,0,0,0,0,18,30] >;`

C36.20D6 in GAP, Magma, Sage, TeX

`C_{36}._{20}D_6`
`% in TeX`

`G:=Group("C36.20D6");`
`// GroupNames label`

`G:=SmallGroup(432,195);`
`// by ID`

`G=gap.SmallGroup(432,195);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,64,254,135,58,6164,662,4037,14118]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=1,b^6=a^18,c^2=a^9,b*a*b^-1=a^19,c*a*c^-1=a^17,c*b*c^-1=a^27*b^5>;`
`// generators/relations`

׿
×
𝔽