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G = C9⋊A4order 108 = 22·33

The semidirect product of C9 and A4 acting via A4/C22=C3

metabelian, soluble, monomial

Aliases: C9⋊A4, C2213- 1+2, (C3×A4).C3, (C2×C18)⋊2C3, C3.A41C3, C3.3(C3×A4), (C2×C6).2C32, SmallGroup(108,19)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C9⋊A4
C1C22C2×C6C3×A4 — C9⋊A4
C22C2×C6 — C9⋊A4
C1C3C9

Generators and relations for C9⋊A4
 G = < a,b,c,d | a9=b2=c2=d3=1, ab=ba, ac=ca, dad-1=a7, dbd-1=bc=cb, dcd-1=b >

3C2
12C3
3C6
4C32
4C9
4C9
3A4
3C18
43- 1+2

Character table of C9⋊A4

 class 123A3B3C3D6A6B9A9B9C9D9E9F18A18B18C18D18E18F
 size 13111212333312121212333333
ρ111111111111111111111    trivial
ρ211111111ζ32ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ31111ζ3ζ321111ζ3ζ32ζ32ζ3111111    linear of order 3
ρ41111ζ32ζ31111ζ32ζ3ζ3ζ32111111    linear of order 3
ρ51111ζ32ζ311ζ32ζ3ζ3ζ3211ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ61111ζ3ζ3211ζ3ζ32ζ32ζ311ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ71111ζ3ζ3211ζ32ζ311ζ3ζ32ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ811111111ζ3ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ91111ζ32ζ311ζ3ζ3211ζ32ζ3ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ103-13300-1-1330000-1-1-1-1-1-1    orthogonal lifted from A4
ρ1133-3-3-3/2-3+3-3/200-3-3-3/2-3+3-3/2000000000000    complex lifted from 3- 1+2
ρ1233-3+3-3/2-3-3-3/200-3+3-3/2-3-3-3/2000000000000    complex lifted from 3- 1+2
ρ133-13300-1-1-3+3-3/2-3-3-3/20000ζ6ζ65ζ65ζ6ζ65ζ6    complex lifted from C3×A4
ρ143-13300-1-1-3-3-3/2-3+3-3/20000ζ65ζ6ζ6ζ65ζ6ζ65    complex lifted from C3×A4
ρ153-1-3-3-3/2-3+3-3/200ζ6ζ6500000098997929495    complex faithful
ρ163-1-3-3-3/2-3+3-3/200ζ6ζ6500000095949989792    complex faithful
ρ173-1-3+3-3/2-3-3-3/200ζ65ζ600000094959899297    complex faithful
ρ183-1-3+3-3/2-3-3-3/200ζ65ζ600000097929594989    complex faithful
ρ193-1-3-3-3/2-3+3-3/200ζ6ζ6500000092979495998    complex faithful
ρ203-1-3+3-3/2-3-3-3/200ζ65ζ600000099892979594    complex faithful

Smallest permutation representation of C9⋊A4
On 36 points
Generators in S36
(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)
(1 27)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 34)(11 35)(12 36)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)
(1 17)(2 18)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(9 16)(19 33)(20 34)(21 35)(22 36)(23 28)(24 29)(25 30)(26 31)(27 32)
(2 5 8)(3 9 6)(10 31 23)(11 35 21)(12 30 19)(13 34 26)(14 29 24)(15 33 22)(16 28 20)(17 32 27)(18 36 25)

G:=sub<Sym(36)| (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), (1,27)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33), (1,17)(2,18)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(19,33)(20,34)(21,35)(22,36)(23,28)(24,29)(25,30)(26,31)(27,32), (2,5,8)(3,9,6)(10,31,23)(11,35,21)(12,30,19)(13,34,26)(14,29,24)(15,33,22)(16,28,20)(17,32,27)(18,36,25)>;

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), (1,27)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33), (1,17)(2,18)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(19,33)(20,34)(21,35)(22,36)(23,28)(24,29)(25,30)(26,31)(27,32), (2,5,8)(3,9,6)(10,31,23)(11,35,21)(12,30,19)(13,34,26)(14,29,24)(15,33,22)(16,28,20)(17,32,27)(18,36,25) );

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)], [(1,27),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,34),(11,35),(12,36),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33)], [(1,17),(2,18),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(9,16),(19,33),(20,34),(21,35),(22,36),(23,28),(24,29),(25,30),(26,31),(27,32)], [(2,5,8),(3,9,6),(10,31,23),(11,35,21),(12,30,19),(13,34,26),(14,29,24),(15,33,22),(16,28,20),(17,32,27),(18,36,25)])

C9⋊A4 is a maximal subgroup of
D9⋊A4  C62.25C32  A4×3- 1+2  C62.9C32  C42⋊3- 1+2  C24⋊3- 1+2  C2423- 1+2  C2443- 1+2
C9⋊A4 is a maximal quotient of
C18.A4  C62.11C32  C62.12C32  C62.16C32  C42⋊3- 1+2  C24⋊3- 1+2  C2423- 1+2  C2443- 1+2

Matrix representation of C9⋊A4 in GL3(𝔽19) generated by

11410
1319
6157
,
001
181818
100
,
010
100
181818
,
100
001
181818
G:=sub<GL(3,GF(19))| [11,13,6,4,1,15,10,9,7],[0,18,1,0,18,0,1,18,0],[0,1,18,1,0,18,0,0,18],[1,0,18,0,0,18,0,1,18] >;

C9⋊A4 in GAP, Magma, Sage, TeX

C_9\rtimes A_4
% in TeX

G:=Group("C9:A4");
// GroupNames label

G:=SmallGroup(108,19);
// by ID

G=gap.SmallGroup(108,19);
# by ID

G:=PCGroup([5,-3,-3,-3,-2,2,121,36,1083,2029]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^7,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

Export

Subgroup lattice of C9⋊A4 in TeX
Character table of C9⋊A4 in TeX

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