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G = Q8⋊C9order 72 = 23·32

The semidirect product of Q8 and C9 acting via C9/C3=C3

non-abelian, soluble

Aliases: Q8⋊C9, C6.1A4, C3.SL2(𝔽3), (C3×Q8).C3, C2.(C3.A4), SmallGroup(72,3)

Series: Derived Chief Lower central Upper central

C1C2Q8 — Q8⋊C9
C1C2Q8C3×Q8 — Q8⋊C9
Q8 — Q8⋊C9
C1C6

Generators and relations for Q8⋊C9
 G = < a,b,c | a4=c9=1, b2=a2, bab-1=a-1, cac-1=b, cbc-1=ab >

3C4
4C9
3C12
4C18

Character table of Q8⋊C9

 class 123A3B46A6B9A9B9C9D9E9F12A12B18A18B18C18D18E18F
 size 111161144444466444444
ρ1111111111111111111111    trivial
ρ21111111ζ32ζ3ζ3ζ3ζ32ζ3211ζ32ζ32ζ3ζ32ζ3ζ3    linear of order 3
ρ31111111ζ3ζ32ζ32ζ32ζ3ζ311ζ3ζ3ζ32ζ3ζ32ζ32    linear of order 3
ρ411ζ32ζ31ζ3ζ32ζ97ζ92ζ98ζ95ζ94ζ9ζ3ζ32ζ97ζ94ζ92ζ9ζ95ζ98    linear of order 9
ρ511ζ3ζ321ζ32ζ3ζ92ζ97ζ9ζ94ζ95ζ98ζ32ζ3ζ92ζ95ζ97ζ98ζ94ζ9    linear of order 9
ρ611ζ3ζ321ζ32ζ3ζ95ζ94ζ97ζ9ζ98ζ92ζ32ζ3ζ95ζ98ζ94ζ92ζ9ζ97    linear of order 9
ρ711ζ32ζ31ζ3ζ32ζ9ζ98ζ95ζ92ζ97ζ94ζ3ζ32ζ9ζ97ζ98ζ94ζ92ζ95    linear of order 9
ρ811ζ3ζ321ζ32ζ3ζ98ζ9ζ94ζ97ζ92ζ95ζ32ζ3ζ98ζ92ζ9ζ95ζ97ζ94    linear of order 9
ρ911ζ32ζ31ζ3ζ32ζ94ζ95ζ92ζ98ζ9ζ97ζ3ζ32ζ94ζ9ζ95ζ97ζ98ζ92    linear of order 9
ρ102-2220-2-2-1-1-1-1-1-100111111    symplectic lifted from SL2(𝔽3), Schur index 2
ρ112-2220-2-2ζ6ζ65ζ65ζ65ζ6ζ600ζ32ζ32ζ3ζ32ζ3ζ3    complex lifted from SL2(𝔽3)
ρ122-2220-2-2ζ65ζ6ζ6ζ6ζ65ζ6500ζ3ζ3ζ32ζ3ζ32ζ32    complex lifted from SL2(𝔽3)
ρ132-2-1+-3-1--301+-31--39297994959800ζ92ζ95ζ97ζ98ζ94ζ9    complex faithful
ρ142-2-1--3-1+-301--31+-39495929899700ζ94ζ9ζ95ζ97ζ98ζ92    complex faithful
ρ152-2-1+-3-1--301+-31--39594979989200ζ95ζ98ζ94ζ92ζ9ζ97    complex faithful
ρ162-2-1+-3-1--301+-31--39899497929500ζ98ζ92ζ9ζ95ζ97ζ94    complex faithful
ρ172-2-1--3-1+-301--31+-39989592979400ζ9ζ97ζ98ζ94ζ92ζ95    complex faithful
ρ182-2-1--3-1+-301--31+-39792989594900ζ97ζ94ζ92ζ9ζ95ζ98    complex faithful
ρ193333-133000000-1-1000000    orthogonal lifted from A4
ρ2033-3+3-3/2-3-3-3/2-1-3-3-3/2-3+3-3/2000000ζ6ζ65000000    complex lifted from C3.A4
ρ2133-3-3-3/2-3+3-3/2-1-3+3-3/2-3-3-3/2000000ζ65ζ6000000    complex lifted from C3.A4

Smallest permutation representation of Q8⋊C9
Regular action on 72 points
Generators in S72
(1 18 46 28)(2 40 47 19)(3 55 48 66)(4 12 49 31)(5 43 50 22)(6 58 51 69)(7 15 52 34)(8 37 53 25)(9 61 54 72)(10 63 29 65)(11 20 30 41)(13 57 32 68)(14 23 33 44)(16 60 35 71)(17 26 36 38)(21 56 42 67)(24 59 45 70)(27 62 39 64)
(1 39 46 27)(2 63 47 65)(3 11 48 30)(4 42 49 21)(5 57 50 68)(6 14 51 33)(7 45 52 24)(8 60 53 71)(9 17 54 36)(10 19 29 40)(12 56 31 67)(13 22 32 43)(15 59 34 70)(16 25 35 37)(18 62 28 64)(20 55 41 66)(23 58 44 69)(26 61 38 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)| (1,18,46,28)(2,40,47,19)(3,55,48,66)(4,12,49,31)(5,43,50,22)(6,58,51,69)(7,15,52,34)(8,37,53,25)(9,61,54,72)(10,63,29,65)(11,20,30,41)(13,57,32,68)(14,23,33,44)(16,60,35,71)(17,26,36,38)(21,56,42,67)(24,59,45,70)(27,62,39,64), (1,39,46,27)(2,63,47,65)(3,11,48,30)(4,42,49,21)(5,57,50,68)(6,14,51,33)(7,45,52,24)(8,60,53,71)(9,17,54,36)(10,19,29,40)(12,56,31,67)(13,22,32,43)(15,59,34,70)(16,25,35,37)(18,62,28,64)(20,55,41,66)(23,58,44,69)(26,61,38,72), (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( (1,18,46,28)(2,40,47,19)(3,55,48,66)(4,12,49,31)(5,43,50,22)(6,58,51,69)(7,15,52,34)(8,37,53,25)(9,61,54,72)(10,63,29,65)(11,20,30,41)(13,57,32,68)(14,23,33,44)(16,60,35,71)(17,26,36,38)(21,56,42,67)(24,59,45,70)(27,62,39,64), (1,39,46,27)(2,63,47,65)(3,11,48,30)(4,42,49,21)(5,57,50,68)(6,14,51,33)(7,45,52,24)(8,60,53,71)(9,17,54,36)(10,19,29,40)(12,56,31,67)(13,22,32,43)(15,59,34,70)(16,25,35,37)(18,62,28,64)(20,55,41,66)(23,58,44,69)(26,61,38,72), (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([(1,18,46,28),(2,40,47,19),(3,55,48,66),(4,12,49,31),(5,43,50,22),(6,58,51,69),(7,15,52,34),(8,37,53,25),(9,61,54,72),(10,63,29,65),(11,20,30,41),(13,57,32,68),(14,23,33,44),(16,60,35,71),(17,26,36,38),(21,56,42,67),(24,59,45,70),(27,62,39,64)], [(1,39,46,27),(2,63,47,65),(3,11,48,30),(4,42,49,21),(5,57,50,68),(6,14,51,33),(7,45,52,24),(8,60,53,71),(9,17,54,36),(10,19,29,40),(12,56,31,67),(13,22,32,43),(15,59,34,70),(16,25,35,37),(18,62,28,64),(20,55,41,66),(23,58,44,69),(26,61,38,72)], [(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⋊C9 is a maximal subgroup of
Q8.D9  Q8⋊D9  Q8.C18  C9×SL2(𝔽3)  C18.A4  Q8⋊3- 1+2  C22⋊(Q8⋊C9)  2+ 1+42C9
Q8⋊C9 is a maximal quotient of
Q8⋊C27  C2.(C42⋊C9)  C22⋊(Q8⋊C9)

Matrix representation of Q8⋊C9 in GL2(𝔽19) generated by

85
611
,
716
412
,
179
016
G:=sub<GL(2,GF(19))| [8,6,5,11],[7,4,16,12],[17,0,9,16] >;

Q8⋊C9 in GAP, Magma, Sage, TeX

Q_8\rtimes C_9
% in TeX

G:=Group("Q8:C9");
// GroupNames label

G:=SmallGroup(72,3);
// by ID

G=gap.SmallGroup(72,3);
# by ID

G:=PCGroup([5,-3,-3,-2,2,-2,15,272,72,543,133,58]);
// Polycyclic

G:=Group<a,b,c|a^4=c^9=1,b^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,c*b*c^-1=a*b>;
// generators/relations

Export

Subgroup lattice of Q8⋊C9 in TeX
Character table of Q8⋊C9 in TeX

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