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G = Q8.C18order 144 = 24·32

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

non-abelian, soluble

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

C1C2Q8 — Q8.C18
C1C2Q8C3×Q8Q8⋊C9 — Q8.C18
Q8 — Q8.C18
C1C12

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 >

6C2
3C4
3C22
6C6
4C9
3D4
3C2×C4
3C12
3C2×C6
4C18
3C2×C12
3C3×D4
4C36

Smallest permutation representation of Q8.C18
On 72 points
Generators in S72
(1 67 19 49)(2 41 20 59)(3 30 21 12)(4 70 22 52)(5 44 23 62)(6 33 24 15)(7 37 25 55)(8 47 26 65)(9 36 27 18)(10 40 28 58)(11 50 29 68)(13 43 31 61)(14 53 32 71)(16 46 34 64)(17 56 35 38)(39 48 57 66)(42 51 60 69)(45 54 63 72)
(1 28 19 10)(2 68 20 50)(3 42 21 60)(4 31 22 13)(5 71 23 53)(6 45 24 63)(7 34 25 16)(8 38 26 56)(9 48 27 66)(11 41 29 59)(12 51 30 69)(14 44 32 62)(15 54 33 72)(17 47 35 65)(18 57 36 39)(37 46 55 64)(40 49 58 67)(43 52 61 70)
(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)| (1,67,19,49)(2,41,20,59)(3,30,21,12)(4,70,22,52)(5,44,23,62)(6,33,24,15)(7,37,25,55)(8,47,26,65)(9,36,27,18)(10,40,28,58)(11,50,29,68)(13,43,31,61)(14,53,32,71)(16,46,34,64)(17,56,35,38)(39,48,57,66)(42,51,60,69)(45,54,63,72), (1,28,19,10)(2,68,20,50)(3,42,21,60)(4,31,22,13)(5,71,23,53)(6,45,24,63)(7,34,25,16)(8,38,26,56)(9,48,27,66)(11,41,29,59)(12,51,30,69)(14,44,32,62)(15,54,33,72)(17,47,35,65)(18,57,36,39)(37,46,55,64)(40,49,58,67)(43,52,61,70), (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( (1,67,19,49)(2,41,20,59)(3,30,21,12)(4,70,22,52)(5,44,23,62)(6,33,24,15)(7,37,25,55)(8,47,26,65)(9,36,27,18)(10,40,28,58)(11,50,29,68)(13,43,31,61)(14,53,32,71)(16,46,34,64)(17,56,35,38)(39,48,57,66)(42,51,60,69)(45,54,63,72), (1,28,19,10)(2,68,20,50)(3,42,21,60)(4,31,22,13)(5,71,23,53)(6,45,24,63)(7,34,25,16)(8,38,26,56)(9,48,27,66)(11,41,29,59)(12,51,30,69)(14,44,32,62)(15,54,33,72)(17,47,35,65)(18,57,36,39)(37,46,55,64)(40,49,58,67)(43,52,61,70), (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([(1,67,19,49),(2,41,20,59),(3,30,21,12),(4,70,22,52),(5,44,23,62),(6,33,24,15),(7,37,25,55),(8,47,26,65),(9,36,27,18),(10,40,28,58),(11,50,29,68),(13,43,31,61),(14,53,32,71),(16,46,34,64),(17,56,35,38),(39,48,57,66),(42,51,60,69),(45,54,63,72)], [(1,28,19,10),(2,68,20,50),(3,42,21,60),(4,31,22,13),(5,71,23,53),(6,45,24,63),(7,34,25,16),(8,38,26,56),(9,48,27,66),(11,41,29,59),(12,51,30,69),(14,44,32,62),(15,54,33,72),(17,47,35,65),(18,57,36,39),(37,46,55,64),(40,49,58,67),(43,52,61,70)], [(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 2A2B3A3B4A4B4C6A6B6C6D9A···9F12A12B12C12D12E12F18A···18F36A···36L
order1223344466669···912121212121218···1836···36
size1161111611664···41111664···44···4

42 irreducible representations

dim111111223333
type++++
imageC1C2C3C6C9C18C4.A4Q8.C18A4C2×A4C3.A4C2×C3.A4
kernelQ8.C18Q8⋊C9C3×C4○D4C3×Q8C4○D4Q8C3C1C12C6C4C2
# reps1122666121122

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

036
10
,
310
06
,
2629
1129
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

Export

Subgroup lattice of Q8.C18 in TeX

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