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## G = C36.69D6order 432 = 24·33

### 19th non-split extension by C36 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — C36.69D6
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C3×C36 — C36.S3 — C36.69D6
 Lower central C3×C9 — C3×C18 — C36.69D6
 Upper central C1 — C4 — C2×C4

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 [×3], C4 [×2], C22, C6, C6 [×3], C6 [×4], C8 [×2], C2×C4, C9 [×3], C32, C12 [×2], C12 [×6], C2×C6, C2×C6 [×3], M4(2), C18 [×3], C18 [×3], C3×C6, C3×C6, C3⋊C8 [×8], C2×C12, C2×C12 [×3], C3×C9, C36 [×6], C2×C18 [×3], C3×C12 [×2], C62, C4.Dic3 [×4], C3×C18, C3×C18, C9⋊C8 [×6], C2×C36 [×3], C324C8 [×2], C6×C12, C3×C36 [×2], C6×C18, C4.Dic9 [×3], C12.58D6, C36.S3 [×2], C6×C36, C36.69D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, Dic3 [×8], D6 [×4], M4(2), D9 [×3], C3⋊S3, C2×Dic3 [×4], Dic9 [×6], D18 [×3], C3⋊Dic3 [×2], C2×C3⋊S3, C4.Dic3 [×4], C9⋊S3, C2×Dic9 [×3], C2×C3⋊Dic3, C9⋊Dic3 [×2], C2×C9⋊S3, C4.Dic9 [×3], C12.58D6, C2×C9⋊Dic3, C36.69D6

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

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

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

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

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`

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