Copied to
clipboard

## G = C36.17D4order 288 = 25·32

### 17th non-split extension by C36 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C36.17D4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — C4×Dic9 — C36.17D4
 Lower central C9 — C2×C18 — C36.17D4
 Upper central C1 — C22 — C2×D4

Generators and relations for C36.17D4
G = < a,b,c | a36=b4=1, c2=a18, bab-1=a17, cac-1=a-1, cbc-1=a18b-1 >

Subgroups: 404 in 114 conjugacy classes, 44 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C9, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C18, C18 [×2], C18 [×2], Dic6 [×2], C2×Dic3 [×4], C2×C12, C3×D4 [×2], C22×C6 [×2], C4.4D4, Dic9 [×4], C36 [×2], C2×C18, C2×C18 [×6], C4×Dic3, C6.D4 [×4], C2×Dic6, C6×D4, Dic18 [×2], C2×Dic9 [×4], C2×C36, D4×C9 [×2], C22×C18 [×2], C23.12D6, C4×Dic9, C18.D4 [×4], C2×Dic18, D4×C18, C36.17D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], D9, C3⋊D4 [×2], C22×S3, C4.4D4, D18 [×3], D42S3 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C23.12D6, D42D9 [×2], C2×C9⋊D4, C36.17D4

Smallest permutation representation of C36.17D4
On 144 points
Generators in S144
```(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)
(1 105 39 142)(2 86 40 123)(3 103 41 140)(4 84 42 121)(5 101 43 138)(6 82 44 119)(7 99 45 136)(8 80 46 117)(9 97 47 134)(10 78 48 115)(11 95 49 132)(12 76 50 113)(13 93 51 130)(14 74 52 111)(15 91 53 128)(16 108 54 109)(17 89 55 126)(18 106 56 143)(19 87 57 124)(20 104 58 141)(21 85 59 122)(22 102 60 139)(23 83 61 120)(24 100 62 137)(25 81 63 118)(26 98 64 135)(27 79 65 116)(28 96 66 133)(29 77 67 114)(30 94 68 131)(31 75 69 112)(32 92 70 129)(33 73 71 110)(34 90 72 127)(35 107 37 144)(36 88 38 125)
(1 133 19 115)(2 132 20 114)(3 131 21 113)(4 130 22 112)(5 129 23 111)(6 128 24 110)(7 127 25 109)(8 126 26 144)(9 125 27 143)(10 124 28 142)(11 123 29 141)(12 122 30 140)(13 121 31 139)(14 120 32 138)(15 119 33 137)(16 118 34 136)(17 117 35 135)(18 116 36 134)(37 98 55 80)(38 97 56 79)(39 96 57 78)(40 95 58 77)(41 94 59 76)(42 93 60 75)(43 92 61 74)(44 91 62 73)(45 90 63 108)(46 89 64 107)(47 88 65 106)(48 87 66 105)(49 86 67 104)(50 85 68 103)(51 84 69 102)(52 83 70 101)(53 82 71 100)(54 81 72 99)```

`G:=sub<Sym(144)| (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), (1,105,39,142)(2,86,40,123)(3,103,41,140)(4,84,42,121)(5,101,43,138)(6,82,44,119)(7,99,45,136)(8,80,46,117)(9,97,47,134)(10,78,48,115)(11,95,49,132)(12,76,50,113)(13,93,51,130)(14,74,52,111)(15,91,53,128)(16,108,54,109)(17,89,55,126)(18,106,56,143)(19,87,57,124)(20,104,58,141)(21,85,59,122)(22,102,60,139)(23,83,61,120)(24,100,62,137)(25,81,63,118)(26,98,64,135)(27,79,65,116)(28,96,66,133)(29,77,67,114)(30,94,68,131)(31,75,69,112)(32,92,70,129)(33,73,71,110)(34,90,72,127)(35,107,37,144)(36,88,38,125), (1,133,19,115)(2,132,20,114)(3,131,21,113)(4,130,22,112)(5,129,23,111)(6,128,24,110)(7,127,25,109)(8,126,26,144)(9,125,27,143)(10,124,28,142)(11,123,29,141)(12,122,30,140)(13,121,31,139)(14,120,32,138)(15,119,33,137)(16,118,34,136)(17,117,35,135)(18,116,36,134)(37,98,55,80)(38,97,56,79)(39,96,57,78)(40,95,58,77)(41,94,59,76)(42,93,60,75)(43,92,61,74)(44,91,62,73)(45,90,63,108)(46,89,64,107)(47,88,65,106)(48,87,66,105)(49,86,67,104)(50,85,68,103)(51,84,69,102)(52,83,70,101)(53,82,71,100)(54,81,72,99)>;`

`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), (1,105,39,142)(2,86,40,123)(3,103,41,140)(4,84,42,121)(5,101,43,138)(6,82,44,119)(7,99,45,136)(8,80,46,117)(9,97,47,134)(10,78,48,115)(11,95,49,132)(12,76,50,113)(13,93,51,130)(14,74,52,111)(15,91,53,128)(16,108,54,109)(17,89,55,126)(18,106,56,143)(19,87,57,124)(20,104,58,141)(21,85,59,122)(22,102,60,139)(23,83,61,120)(24,100,62,137)(25,81,63,118)(26,98,64,135)(27,79,65,116)(28,96,66,133)(29,77,67,114)(30,94,68,131)(31,75,69,112)(32,92,70,129)(33,73,71,110)(34,90,72,127)(35,107,37,144)(36,88,38,125), (1,133,19,115)(2,132,20,114)(3,131,21,113)(4,130,22,112)(5,129,23,111)(6,128,24,110)(7,127,25,109)(8,126,26,144)(9,125,27,143)(10,124,28,142)(11,123,29,141)(12,122,30,140)(13,121,31,139)(14,120,32,138)(15,119,33,137)(16,118,34,136)(17,117,35,135)(18,116,36,134)(37,98,55,80)(38,97,56,79)(39,96,57,78)(40,95,58,77)(41,94,59,76)(42,93,60,75)(43,92,61,74)(44,91,62,73)(45,90,63,108)(46,89,64,107)(47,88,65,106)(48,87,66,105)(49,86,67,104)(50,85,68,103)(51,84,69,102)(52,83,70,101)(53,82,71,100)(54,81,72,99) );`

`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)], [(1,105,39,142),(2,86,40,123),(3,103,41,140),(4,84,42,121),(5,101,43,138),(6,82,44,119),(7,99,45,136),(8,80,46,117),(9,97,47,134),(10,78,48,115),(11,95,49,132),(12,76,50,113),(13,93,51,130),(14,74,52,111),(15,91,53,128),(16,108,54,109),(17,89,55,126),(18,106,56,143),(19,87,57,124),(20,104,58,141),(21,85,59,122),(22,102,60,139),(23,83,61,120),(24,100,62,137),(25,81,63,118),(26,98,64,135),(27,79,65,116),(28,96,66,133),(29,77,67,114),(30,94,68,131),(31,75,69,112),(32,92,70,129),(33,73,71,110),(34,90,72,127),(35,107,37,144),(36,88,38,125)], [(1,133,19,115),(2,132,20,114),(3,131,21,113),(4,130,22,112),(5,129,23,111),(6,128,24,110),(7,127,25,109),(8,126,26,144),(9,125,27,143),(10,124,28,142),(11,123,29,141),(12,122,30,140),(13,121,31,139),(14,120,32,138),(15,119,33,137),(16,118,34,136),(17,117,35,135),(18,116,36,134),(37,98,55,80),(38,97,56,79),(39,96,57,78),(40,95,58,77),(41,94,59,76),(42,93,60,75),(43,92,61,74),(44,91,62,73),(45,90,63,108),(46,89,64,107),(47,88,65,106),(48,87,66,105),(49,86,67,104),(50,85,68,103),(51,84,69,102),(52,83,70,101),(53,82,71,100),(54,81,72,99)])`

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 12A 12B 18A ··· 18I 18J ··· 18U 36A ··· 36F order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 9 9 9 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 4 4 2 2 2 18 18 18 18 36 36 2 2 2 4 4 4 4 2 2 2 4 4 2 ··· 2 4 ··· 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 D9 C3⋊D4 D18 D18 C9⋊D4 D4⋊2S3 D4⋊2D9 kernel C36.17D4 C4×Dic9 C18.D4 C2×Dic18 D4×C18 C6×D4 C36 C2×C12 C22×C6 C18 C2×D4 C12 C2×C4 C23 C4 C6 C2 # reps 1 1 4 1 1 1 2 1 2 4 3 4 3 6 12 2 6

Matrix representation of C36.17D4 in GL4(𝔽37) generated by

 0 36 0 0 1 0 0 0 0 0 25 26 0 0 0 3
,
 0 6 0 0 31 0 0 0 0 0 6 0 0 0 12 31
,
 31 0 0 0 0 6 0 0 0 0 6 31 0 0 12 31
`G:=sub<GL(4,GF(37))| [0,1,0,0,36,0,0,0,0,0,25,0,0,0,26,3],[0,31,0,0,6,0,0,0,0,0,6,12,0,0,0,31],[31,0,0,0,0,6,0,0,0,0,6,12,0,0,31,31] >;`

C36.17D4 in GAP, Magma, Sage, TeX

`C_{36}._{17}D_4`
`% in TeX`

`G:=Group("C36.17D4");`
`// GroupNames label`

`G:=SmallGroup(288,146);`
`// by ID`

`G=gap.SmallGroup(288,146);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,135,6725,292,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=b^4=1,c^2=a^18,b*a*b^-1=a^17,c*a*c^-1=a^-1,c*b*c^-1=a^18*b^-1>;`
`// generators/relations`

׿
×
𝔽