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## G = C6.11D36order 432 = 24·33

### 11st non-split extension by C6 of D36 acting via D36/C36=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — C6.11D36
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C6×C18 — C22×C9⋊S3 — C6.11D36
 Lower central C3×C9 — C3×C18 — C6.11D36
 Upper central C1 — C22 — C2×C4

Generators and relations for C6.11D36
G = < a,b,c | a18=b12=1, c2=a9, ab=ba, cac-1=a-1, cbc-1=a9b-1 >

Subgroups: 1220 in 170 conjugacy classes, 65 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C3, C3 [×3], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×3], C6 [×9], C2×C4, C2×C4, C23, C9 [×3], C32, Dic3 [×4], C12 [×4], D6 [×16], C2×C6, C2×C6 [×3], C22⋊C4, D9 [×6], C18 [×9], C3⋊S3 [×2], C3×C6 [×3], C2×Dic3 [×4], C2×C12, C2×C12 [×3], C22×S3 [×4], C3×C9, Dic9 [×3], C36 [×3], D18 [×12], C2×C18 [×3], C3⋊Dic3, C3×C12, C2×C3⋊S3 [×4], C62, D6⋊C4 [×4], C9⋊S3 [×2], C3×C18 [×3], C2×Dic9 [×3], C2×C36 [×3], C22×D9 [×3], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C9⋊Dic3, C3×C36, C2×C9⋊S3 [×2], C2×C9⋊S3 [×2], C6×C18, D18⋊C4 [×3], C6.11D12, C2×C9⋊Dic3, C6×C36, C22×C9⋊S3, C6.11D36
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], D6 [×4], C22⋊C4, D9 [×3], C3⋊S3, C4×S3 [×4], D12 [×4], C3⋊D4 [×4], D18 [×3], C2×C3⋊S3, D6⋊C4 [×4], C9⋊S3, C4×D9 [×3], D36 [×3], C9⋊D4 [×3], C4×C3⋊S3, C12⋊S3, C327D4, C2×C9⋊S3, D18⋊C4 [×3], C6.11D12, C4×C9⋊S3, C36⋊S3, C6.D18, C6.11D36

Smallest permutation representation of C6.11D36
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 61 178 38 30 102 137 151 117 85 192 201)(2 62 179 39 31 103 138 152 118 86 193 202)(3 63 180 40 32 104 139 153 119 87 194 203)(4 64 163 41 33 105 140 154 120 88 195 204)(5 65 164 42 34 106 141 155 121 89 196 205)(6 66 165 43 35 107 142 156 122 90 197 206)(7 67 166 44 36 108 143 157 123 73 198 207)(8 68 167 45 19 91 144 158 124 74 181 208)(9 69 168 46 20 92 127 159 125 75 182 209)(10 70 169 47 21 93 128 160 126 76 183 210)(11 71 170 48 22 94 129 161 109 77 184 211)(12 72 171 49 23 95 130 162 110 78 185 212)(13 55 172 50 24 96 131 145 111 79 186 213)(14 56 173 51 25 97 132 146 112 80 187 214)(15 57 174 52 26 98 133 147 113 81 188 215)(16 58 175 53 27 99 134 148 114 82 189 216)(17 59 176 54 28 100 135 149 115 83 190 199)(18 60 177 37 29 101 136 150 116 84 191 200)
(1 93 10 102)(2 92 11 101)(3 91 12 100)(4 108 13 99)(5 107 14 98)(6 106 15 97)(7 105 16 96)(8 104 17 95)(9 103 18 94)(19 63 28 72)(20 62 29 71)(21 61 30 70)(22 60 31 69)(23 59 32 68)(24 58 33 67)(25 57 34 66)(26 56 35 65)(27 55 36 64)(37 179 46 170)(38 178 47 169)(39 177 48 168)(40 176 49 167)(41 175 50 166)(42 174 51 165)(43 173 52 164)(44 172 53 163)(45 171 54 180)(73 111 82 120)(74 110 83 119)(75 109 84 118)(76 126 85 117)(77 125 86 116)(78 124 87 115)(79 123 88 114)(80 122 89 113)(81 121 90 112)(127 202 136 211)(128 201 137 210)(129 200 138 209)(130 199 139 208)(131 216 140 207)(132 215 141 206)(133 214 142 205)(134 213 143 204)(135 212 144 203)(145 198 154 189)(146 197 155 188)(147 196 156 187)(148 195 157 186)(149 194 158 185)(150 193 159 184)(151 192 160 183)(152 191 161 182)(153 190 162 181)```

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

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

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

114 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6L 9A ··· 9I 12A ··· 12P 18A ··· 18AA 36A ··· 36AJ order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 54 54 2 2 2 2 2 2 54 54 2 ··· 2 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 C2 C4 S3 S3 D4 D6 D6 D9 C4×S3 D12 C3⋊D4 C4×S3 D12 C3⋊D4 D18 C4×D9 D36 C9⋊D4 kernel C6.11D36 C2×C9⋊Dic3 C6×C36 C22×C9⋊S3 C2×C9⋊S3 C2×C36 C6×C12 C3×C18 C2×C18 C62 C2×C12 C18 C18 C18 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 # reps 1 1 1 1 4 3 1 2 3 1 9 6 6 6 2 2 2 9 18 18 18

Matrix representation of C6.11D36 in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 20 11 0 0 26 31
,
 5 32 0 0 5 10 0 0 0 0 23 7 0 0 30 30
,
 32 5 0 0 10 5 0 0 0 0 30 30 0 0 23 7
`G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,20,26,0,0,11,31],[5,5,0,0,32,10,0,0,0,0,23,30,0,0,7,30],[32,10,0,0,5,5,0,0,0,0,30,23,0,0,30,7] >;`

C6.11D36 in GAP, Magma, Sage, TeX

`C_6._{11}D_{36}`
`% in TeX`

`G:=Group("C6.11D36");`
`// GroupNames label`

`G:=SmallGroup(432,183);`
`// by ID`

`G=gap.SmallGroup(432,183);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,6164,662,4037,14118]);`
`// Polycyclic`

`G:=Group<a,b,c|a^18=b^12=1,c^2=a^9,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^9*b^-1>;`
`// generators/relations`

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