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G = C922C3order 243 = 35

2nd semidirect product of C92 and C3 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C922C3, C32.2He3, He3⋊C31C3, (C3×C9).17C32, C3.7(He3⋊C3), 3-Sylow(SL(3,19)), SmallGroup(243,26)

Series: Derived Chief Lower central Upper central Jennings

C1C3×C9 — C922C3
C1C3C32C3×C9C92 — C922C3
C1C3C32C3×C9 — C922C3
C1C3C32C3×C9 — C922C3
C1C3C3C3C32C3×C9 — C922C3

Generators and relations for C922C3
 G = < a,b,c | a9=b9=c3=1, ab=ba, cac-1=a7b-1, cbc-1=a3b >

3C3
27C3
27C3
27C3
3C9
3C9
3C9
3C9
9C32
9C32
9C32
3He3
3He3
3C3×C9
3He3

Permutation representations of C922C3
On 27 points - transitive group 27T104
Generators in S27
(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)
(1 8 5 2 9 6 3 7 4)(10 17 15 13 11 18 16 14 12)(19 27 26 25 24 23 22 21 20)
(1 23 10)(2 20 13)(3 26 16)(4 24 18)(5 21 12)(6 27 15)(7 25 17)(8 22 11)(9 19 14)

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

G:=Group( (10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,8,5,2,9,6,3,7,4)(10,17,15,13,11,18,16,14,12)(19,27,26,25,24,23,22,21,20), (1,23,10)(2,20,13)(3,26,16)(4,24,18)(5,21,12)(6,27,15)(7,25,17)(8,22,11)(9,19,14) );

G=PermutationGroup([(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27)], [(1,8,5,2,9,6,3,7,4),(10,17,15,13,11,18,16,14,12),(19,27,26,25,24,23,22,21,20)], [(1,23,10),(2,20,13),(3,26,16),(4,24,18),(5,21,12),(6,27,15),(7,25,17),(8,22,11),(9,19,14)])

G:=TransitiveGroup(27,104);

C922C3 is a maximal subgroup of   C92⋊S3  C922C6  C922S3

35 conjugacy classes

class 1 3A3B3C3D3E···3J9A···9X
order133333···39···9
size1113327···273···3

35 irreducible representations

dim111333
type+
imageC1C3C3He3He3⋊C3C922C3
kernelC922C3C92He3⋊C3C32C3C1
# reps1262618

Matrix representation of C922C3 in GL3(𝔽19) generated by

700
0160
009
,
500
0160
005
,
010
001
100
G:=sub<GL(3,GF(19))| [7,0,0,0,16,0,0,0,9],[5,0,0,0,16,0,0,0,5],[0,0,1,1,0,0,0,1,0] >;

C922C3 in GAP, Magma, Sage, TeX

C_9^2\rtimes_2C_3
% in TeX

G:=Group("C9^2:2C3");
// GroupNames label

G:=SmallGroup(243,26);
// by ID

G=gap.SmallGroup(243,26);
# by ID

G:=PCGroup([5,-3,3,-3,-3,-3,121,456,542,282,2163]);
// Polycyclic

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

Export

Subgroup lattice of C922C3 in TeX

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