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G = C928C6order 486 = 2·35

8th semidirect product of C92 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: C928C6, D923- 1+2, C9⋊C93C6, C9⋊C183C3, C94(C9⋊C6), (C9×D9)⋊4C3, C929C31C2, C33.13(C3×S3), (C3×D9).6C32, C92(C2×3- 1+2), C32.43(S3×C32), C3.6(S3×3- 1+2), (C3×3- 1+2).2C6, (C3×3- 1+2).12S3, C3.8(C3×C9⋊C6), (C3×C9⋊C6).2C3, (C3×C9).12(C3×S3), (C3×C9).15(C3×C6), SmallGroup(486,110)

Series: Derived Chief Lower central Upper central

C1C3×C9 — C928C6
C1C3C9C3×C9C92C929C3 — C928C6
C9C3×C9 — C928C6
C1C3C9

Generators and relations for C928C6
 G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a7, cbc-1=b2 >

Subgroups: 256 in 71 conjugacy classes, 24 normal (18 characteristic)
C1, C2, C3 [×2], C3 [×2], S3, C6 [×2], C9 [×2], C9 [×10], C32, C32 [×3], D9, C18 [×3], C3×S3 [×2], C3×C6, C3×C9 [×2], C3×C9 [×2], C3×C9 [×4], 3- 1+2 [×7], C33, C3×D9, S3×C9 [×3], C9⋊C6, C2×3- 1+2, S3×C32, C92, C9⋊C9 [×2], C9⋊C9 [×3], C3×3- 1+2 [×2], C3×3- 1+2, C9×D9, C9⋊C18 [×2], C3×C9⋊C6, S3×3- 1+2, C929C3, C928C6
Quotients: C1, C2, C3 [×4], S3, C6 [×4], C32, C3×S3 [×4], C3×C6, 3- 1+2, C9⋊C6, C2×3- 1+2, S3×C32, C3×C9⋊C6, S3×3- 1+2, C928C6

Permutation representations of C928C6
On 18 points - transitive group 18T158
Generators in S18
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 9 8 7 6 5 4 3 2)(10 11 12 13 14 15 16 17 18)
(1 16)(2 11 8 17 5 14)(3 15 6 18 9 12)(4 10)(7 13)

G:=sub<Sym(18)| (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,9,8,7,6,5,4,3,2)(10,11,12,13,14,15,16,17,18), (1,16)(2,11,8,17,5,14)(3,15,6,18,9,12)(4,10)(7,13)>;

G:=Group( (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,9,8,7,6,5,4,3,2)(10,11,12,13,14,15,16,17,18), (1,16)(2,11,8,17,5,14)(3,15,6,18,9,12)(4,10)(7,13) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,9,8,7,6,5,4,3,2),(10,11,12,13,14,15,16,17,18)], [(1,16),(2,11,8,17,5,14),(3,15,6,18,9,12),(4,10),(7,13)])

G:=TransitiveGroup(18,158);

42 conjugacy classes

class 1  2 3A3B3C3D3E3F3G6A6B6C6D9A9B9C···9M9N9O9P9Q9R···9W18A···18F
order1233333336666999···999999···918···18
size191122299992727336···6999918···1827···27

42 irreducible representations

dim11111111222336666
type++++
imageC1C2C3C3C3C6C6C6S3C3×S3C3×S33- 1+2C2×3- 1+2C9⋊C6C3×C9⋊C6S3×3- 1+2C928C6
kernelC928C6C929C3C9×D9C9⋊C18C3×C9⋊C6C92C9⋊C9C3×3- 1+2C3×3- 1+2C3×C9C33D9C9C9C3C3C1
# reps11242242162221226

Matrix representation of C928C6 in GL6(𝔽19)

106000
8018000
81118000
001001
11011100
11010110
,
740000
0127000
18120000
170001
12701100
12700110
,
1800600
18001870
110018011
000100
170100
8011100

G:=sub<GL(6,GF(19))| [1,8,8,0,11,11,0,0,11,0,0,0,6,18,18,1,1,1,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[7,0,18,1,12,12,4,12,12,7,7,7,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[18,18,11,0,1,8,0,0,0,0,7,0,0,0,0,0,0,11,6,18,18,1,1,1,0,7,0,0,0,0,0,0,11,0,0,0] >;

C928C6 in GAP, Magma, Sage, TeX

C_9^2\rtimes_8C_6
% in TeX

G:=Group("C9^2:8C6");
// GroupNames label

G:=SmallGroup(486,110);
// by ID

G=gap.SmallGroup(486,110);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,122,8104,3250,208,11669]);
// Polycyclic

G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,c*a*c^-1=a^7,c*b*c^-1=b^2>;
// generators/relations

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