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G = C92⋊C4order 324 = 22·34

The semidirect product of C92 and C4 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C92⋊C4, C9⋊D9.C2, C32.(C32⋊C4), SmallGroup(324,35)

Series: Derived Chief Lower central Upper central

C1C92 — C92⋊C4
C1C32C92C9⋊D9 — C92⋊C4
C92 — C92⋊C4
C1

Generators and relations for C92⋊C4
 G = < a,b,c | a9=b9=c4=1, ab=ba, cac-1=a-1b4, cbc-1=a4b >

81C2
2C3
2C3
81C4
54S3
54S3
2C9
2C9
2C9
2C9
2C9
2C9
9C3⋊S3
18D9
18D9
18D9
18D9
18D9
18D9
2C3×C9
2C3×C9
9C32⋊C4
6C9⋊S3
6C9⋊S3

Character table of C92⋊C4

 class 123A3B4A4B9A9B9C9D9E9F9G9H9I9J9K9L9M9N9O9P9Q9R
 size 181448181444444444444444444
ρ1111111111111111111111111    trivial
ρ21111-1-1111111111111111111    linear of order 2
ρ31-111-ii111111111111111111    linear of order 4
ρ41-111i-i111111111111111111    linear of order 4
ρ5404400111111-2-2-2-2-2-2-2-2-2111    orthogonal lifted from C32⋊C4
ρ6404400-2-2-2-2-2-2111111111-2-2-2    orthogonal lifted from C32⋊C4
ρ740-2100ζ989+2ζ9792+2ζ9594+2ζ9594-1ζ989-1ζ9792-1989959497+2ζ929594979298+2ζ9979298995+2ζ94ζ9792-1ζ9594-1ζ989-1    orthogonal faithful
ρ8401-20098+2ζ997+2ζ9295+2ζ9495949899792ζ9792-1ζ989-1ζ9594+2ζ989-1ζ9594-1ζ9792+2ζ9594-1ζ9792-1ζ989+297929594989    orthogonal faithful
ρ9401-2009594989979297929594989ζ989+2ζ9594-1ζ9792-1ζ9594+2ζ9792-1ζ989-1ζ9792+2ζ989-1ζ9594-198+2ζ997+2ζ9295+2ζ94    orthogonal faithful
ρ1040-2100ζ9792+2ζ9594+2ζ989+2ζ989-1ζ9792-1ζ9594-1979298995+2ζ94989959497+2ζ929594979298+2ζ9ζ9594-1ζ989-1ζ9792-1    orthogonal faithful
ρ1140-2100ζ9792-1ζ9594-1ζ989-1ζ989+2ζ9792+2ζ9594+2979298+2ζ9959498995+2ζ949792959497+2ζ92989ζ9594-1ζ989-1ζ9792-1    orthogonal faithful
ρ12401-2009899792959495949899792ζ9792+2ζ989-1ζ9594-1ζ989+2ζ9594-1ζ9792-1ζ9594+2ζ9792-1ζ989-197+2ζ9295+2ζ9498+2ζ9    orthogonal faithful
ρ1340-2100ζ9792-1ζ9594-1ζ989-1ζ989-1ζ9792-1ζ9594-197+2ζ92989959498+2ζ99594979295+2ζ949792989ζ9594+2ζ989+2ζ9792+2    orthogonal faithful
ρ1440-2100ζ9594-1ζ989-1ζ9792-1ζ9792-1ζ9594-1ζ989-195+2ζ94979298997+2ζ92989959498+2ζ995949792ζ989+2ζ9792+2ζ9594+2    orthogonal faithful
ρ15401-2009899792959495+2ζ9498+2ζ997+2ζ92ζ9792-1ζ989+2ζ9594-1ζ989-1ζ9594+2ζ9792-1ζ9594-1ζ9792+2ζ989-197929594989    orthogonal faithful
ρ16401-20097+2ζ9295+2ζ9498+2ζ998997929594ζ9594-1ζ9792-1ζ989+2ζ9792-1ζ989-1ζ9594+2ζ989-1ζ9594-1ζ9792+295949899792    orthogonal faithful
ρ17401-20095+2ζ9498+2ζ997+2ζ9297929594989ζ989-1ζ9594-1ζ9792+2ζ9594-1ζ9792-1ζ989+2ζ9792-1ζ989-1ζ9594+298997929594    orthogonal faithful
ρ1840-2100ζ989-1ζ9792-1ζ9594-1ζ9594+2ζ989+2ζ9792+298995+2ζ949792959497+2ζ92989979298+2ζ99594ζ9792-1ζ9594-1ζ989-1    orthogonal faithful
ρ19401-2009792959498998997929594ζ9594+2ζ9792-1ζ989-1ζ9792+2ζ989-1ζ9594-1ζ989+2ζ9594-1ζ9792-195+2ζ9498+2ζ997+2ζ92    orthogonal faithful
ρ2040-2100ζ9594+2ζ989+2ζ9792+2ζ9792-1ζ9594-1ζ989-19594979298+2ζ9979298995+2ζ94989959497+2ζ92ζ989-1ζ9792-1ζ9594-1    orthogonal faithful
ρ2140-2100ζ9594-1ζ989-1ζ9792-1ζ9792+2ζ9594+2ζ989+2959497+2ζ92989979298+2ζ9959498995+2ζ949792ζ989-1ζ9792-1ζ9594-1    orthogonal faithful
ρ22401-2009594989979297+2ζ9295+2ζ9498+2ζ9ζ989-1ζ9594+2ζ9792-1ζ9594-1ζ9792+2ζ989-1ζ9792-1ζ989+2ζ9594-198997929594    orthogonal faithful
ρ2340-2100ζ989-1ζ9792-1ζ9594-1ζ9594-1ζ989-1ζ9792-198+2ζ99594979295+2ζ94979298997+2ζ929899594ζ9792+2ζ9594+2ζ989+2    orthogonal faithful
ρ24401-2009792959498998+2ζ997+2ζ9295+2ζ94ζ9594-1ζ9792+2ζ989-1ζ9792-1ζ989+2ζ9594-1ζ989-1ζ9594+2ζ9792-195949899792    orthogonal faithful

Permutation representations of C92⋊C4
On 18 points - transitive group 18T130
Generators in S18
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 4 7)(2 5 8)(3 6 9)(10 18 17 16 15 14 13 12 11)
(1 17)(2 13 9 12)(3 18 8 16)(4 14 7 11)(5 10 6 15)

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

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

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

G:=TransitiveGroup(18,130);

Matrix representation of C92⋊C4 in GL4(𝔽37) generated by

61100
261700
001120
001731
,
36100
36000
001726
00116
,
0010
0001
20600
261700
G:=sub<GL(4,GF(37))| [6,26,0,0,11,17,0,0,0,0,11,17,0,0,20,31],[36,36,0,0,1,0,0,0,0,0,17,11,0,0,26,6],[0,0,20,26,0,0,6,17,1,0,0,0,0,1,0,0] >;

C92⋊C4 in GAP, Magma, Sage, TeX

C_9^2\rtimes C_4
% in TeX

G:=Group("C9^2:C4");
// GroupNames label

G:=SmallGroup(324,35);
// by ID

G=gap.SmallGroup(324,35);
# by ID

G:=PCGroup([6,-2,-2,-3,3,-3,3,12,506,404,338,6819,2889,237,4324,1090,9077,3899]);
// Polycyclic

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

Export

Subgroup lattice of C92⋊C4 in TeX
Character table of C92⋊C4 in TeX

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