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## G = Q8.C18order 144 = 24·32

### The non-split extension by Q8 of C18 acting via C18/C6=C3

Aliases: Q8.C18, C12.2A4, C4○D4⋊C9, Q8⋊C92C2, C6.6(C2×A4), C3.(C4.A4), C4.(C3.A4), (C3×Q8).3C6, (C3×C4○D4).C3, C2.3(C2×C3.A4), SmallGroup(144,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — Q8.C18
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8⋊C9 — Q8.C18
 Lower central Q8 — Q8.C18
 Upper central C1 — C12

Generators and relations for Q8.C18
G = < a,b,c | a4=1, b2=c18=a2, bab-1=a-1, cac-1=ab, cbc-1=a >

Smallest permutation representation of Q8.C18
On 72 points
Generators in S72
```(1 42 19 60)(2 52 20 70)(3 30 21 12)(4 45 22 63)(5 55 23 37)(6 33 24 15)(7 48 25 66)(8 58 26 40)(9 36 27 18)(10 51 28 69)(11 61 29 43)(13 54 31 72)(14 64 32 46)(16 57 34 39)(17 67 35 49)(38 47 56 65)(41 50 59 68)(44 53 62 71)
(1 28 19 10)(2 43 20 61)(3 53 21 71)(4 31 22 13)(5 46 23 64)(6 56 24 38)(7 34 25 16)(8 49 26 67)(9 59 27 41)(11 52 29 70)(12 62 30 44)(14 55 32 37)(15 65 33 47)(17 58 35 40)(18 68 36 50)(39 48 57 66)(42 51 60 69)(45 54 63 72)
(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)```

`G:=sub<Sym(72)| (1,42,19,60)(2,52,20,70)(3,30,21,12)(4,45,22,63)(5,55,23,37)(6,33,24,15)(7,48,25,66)(8,58,26,40)(9,36,27,18)(10,51,28,69)(11,61,29,43)(13,54,31,72)(14,64,32,46)(16,57,34,39)(17,67,35,49)(38,47,56,65)(41,50,59,68)(44,53,62,71), (1,28,19,10)(2,43,20,61)(3,53,21,71)(4,31,22,13)(5,46,23,64)(6,56,24,38)(7,34,25,16)(8,49,26,67)(9,59,27,41)(11,52,29,70)(12,62,30,44)(14,55,32,37)(15,65,33,47)(17,58,35,40)(18,68,36,50)(39,48,57,66)(42,51,60,69)(45,54,63,72), (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)>;`

`G:=Group( (1,42,19,60)(2,52,20,70)(3,30,21,12)(4,45,22,63)(5,55,23,37)(6,33,24,15)(7,48,25,66)(8,58,26,40)(9,36,27,18)(10,51,28,69)(11,61,29,43)(13,54,31,72)(14,64,32,46)(16,57,34,39)(17,67,35,49)(38,47,56,65)(41,50,59,68)(44,53,62,71), (1,28,19,10)(2,43,20,61)(3,53,21,71)(4,31,22,13)(5,46,23,64)(6,56,24,38)(7,34,25,16)(8,49,26,67)(9,59,27,41)(11,52,29,70)(12,62,30,44)(14,55,32,37)(15,65,33,47)(17,58,35,40)(18,68,36,50)(39,48,57,66)(42,51,60,69)(45,54,63,72), (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) );`

`G=PermutationGroup([[(1,42,19,60),(2,52,20,70),(3,30,21,12),(4,45,22,63),(5,55,23,37),(6,33,24,15),(7,48,25,66),(8,58,26,40),(9,36,27,18),(10,51,28,69),(11,61,29,43),(13,54,31,72),(14,64,32,46),(16,57,34,39),(17,67,35,49),(38,47,56,65),(41,50,59,68),(44,53,62,71)], [(1,28,19,10),(2,43,20,61),(3,53,21,71),(4,31,22,13),(5,46,23,64),(6,56,24,38),(7,34,25,16),(8,49,26,67),(9,59,27,41),(11,52,29,70),(12,62,30,44),(14,55,32,37),(15,65,33,47),(17,58,35,40),(18,68,36,50),(39,48,57,66),(42,51,60,69),(45,54,63,72)], [(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)]])`

Q8.C18 is a maximal subgroup of
C12.9S4  Q8.C36  C12.3S4  C12.11S4  C12.4S4  2+ 1+4⋊C9  2- 1+4⋊C9  Q8⋊C93S3  C9×C4.A4  C36.A4  Q8⋊C94C6
Q8.C18 is a maximal quotient of
C4×Q8⋊C9  Q8.C54  Q8⋊C93S3

42 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 6C 6D 9A ··· 9F 12A 12B 12C 12D 12E 12F 18A ··· 18F 36A ··· 36L order 1 2 2 3 3 4 4 4 6 6 6 6 9 ··· 9 12 12 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 6 1 1 1 1 6 1 1 6 6 4 ··· 4 1 1 1 1 6 6 4 ··· 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 3 3 3 3 type + + + + image C1 C2 C3 C6 C9 C18 C4.A4 Q8.C18 A4 C2×A4 C3.A4 C2×C3.A4 kernel Q8.C18 Q8⋊C9 C3×C4○D4 C3×Q8 C4○D4 Q8 C3 C1 C12 C6 C4 C2 # reps 1 1 2 2 6 6 6 12 1 1 2 2

Matrix representation of Q8.C18 in GL2(𝔽37) generated by

 0 36 1 0
,
 31 0 0 6
,
 26 29 11 29
`G:=sub<GL(2,GF(37))| [0,1,36,0],[31,0,0,6],[26,11,29,29] >;`

Q8.C18 in GAP, Magma, Sage, TeX

`Q_8.C_{18}`
`% in TeX`

`G:=Group("Q8.C18");`
`// GroupNames label`

`G:=SmallGroup(144,36);`
`// by ID`

`G=gap.SmallGroup(144,36);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-2,2,-2,432,43,441,117,820,202,88]);`
`// Polycyclic`

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

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