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G = C32.5He3order 243 = 35

5th non-split extension by C32 of He3 acting via He3/C32=C3

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

Aliases: C32.5He3, C9⋊C9.2C3, (C3×C9).3C32, C3.He3.2C3, C3.10(He3⋊C3), SmallGroup(243,29)

Series: Derived Chief Lower central Upper central Jennings

C1C3×C9 — C32.5He3
C1C3C32C3×C9C9⋊C9 — C32.5He3
C1C3C32C3×C9 — C32.5He3
C1C3C32C3×C9 — C32.5He3
C1C3C3C3C32C3×C9 — C32.5He3

Generators and relations for C32.5He3
 G = < a,b,c,d,e | a3=b3=1, c3=a-1, d3=b-1, e3=b, ab=ba, ac=ca, ad=da, eae-1=ab-1, dcd-1=bc=cb, bd=db, be=eb, ece-1=acd2, ede-1=a-1bd >

3C3
3C9
9C9
9C9
9C9
9C9
33- 1+2
33- 1+2
33- 1+2
3C3×C9

Character table of C32.5He3

 class 13A3B3C3D9A9B9C9D9E9F9G9H9I9J9K9L9M9N
 size 1113399999999272727272727
ρ11111111111111111111    trivial
ρ211111ζ311ζ32ζ32ζ3ζ3ζ3211ζ32ζ32ζ3ζ3    linear of order 3
ρ311111ζ311ζ32ζ32ζ3ζ3ζ32ζ32ζ31ζ31ζ32    linear of order 3
ρ411111ζ3211ζ3ζ3ζ32ζ32ζ3ζ3ζ321ζ321ζ3    linear of order 3
ρ51111111111111ζ32ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ61111111111111ζ3ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ711111ζ311ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ31ζ321    linear of order 3
ρ811111ζ3211ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ321ζ31    linear of order 3
ρ911111ζ3211ζ3ζ3ζ32ζ32ζ311ζ3ζ3ζ32ζ32    linear of order 3
ρ10333330-3+3-3/2-3-3-3/200000000000    complex lifted from He3
ρ11333330-3-3-3/2-3+3-3/200000000000    complex lifted from He3
ρ12333-3+3-3/2-3-3-3/2ζ95+2ζ920097949499892ζ98+2ζ95ζ97+2ζ9000000    complex lifted from He3⋊C3
ρ13333-3+3-3/2-3-3-3/2989200ζ97+2ζ99794ζ98+2ζ95ζ95+2ζ92949000000    complex lifted from He3⋊C3
ρ14333-3+3-3/2-3-3-3/2ζ98+2ζ9500949ζ97+2ζ9ζ95+2ζ9298929794000000    complex lifted from He3⋊C3
ρ15333-3-3-3/2-3+3-3/2979400ζ95+2ζ92ζ98+2ζ95ζ97+2ζ99499892000000    complex lifted from He3⋊C3
ρ16333-3-3-3/2-3+3-3/294900ζ98+2ζ9598929794ζ97+2ζ9ζ95+2ζ92000000    complex lifted from He3⋊C3
ρ17333-3-3-3/2-3+3-3/2ζ97+2ζ9009892ζ95+2ζ929499794ζ98+2ζ95000000    complex lifted from He3⋊C3
ρ189-9+9-3/2-9-9-3/20000000000000000    complex faithful
ρ199-9-9-3/2-9+9-3/20000000000000000    complex faithful

Permutation representations of C32.5He3
On 27 points - transitive group 27T90
Generators in S27
(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)
(1 3 2)(4 6 5)(7 8 9)(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)
(4 5 6)(7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)
(1 7 5 2 9 6 3 8 4)(10 14 12 13 17 15 16 11 18)(19 27 20 25 24 26 22 21 23)
(1 21 15 3 24 12 2 27 18)(4 25 17 6 19 14 5 22 11)(7 20 16 8 23 13 9 26 10)

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

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

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

G:=TransitiveGroup(27,90);

C32.5He3 is a maximal subgroup of   C9⋊C9.3S3

Matrix representation of C32.5He3 in GL9(𝔽19)

100000000
010000000
001000000
000700000
000070000
000007000
0000001100
0000000110
0000000011
,
700000000
070000000
007000000
000700000
000070000
000007000
000000700
000000070
000000007
,
100000000
0110000000
8117000000
0000110000
0001884000
0007111000
0000001884
0000001100
0000001101
,
010000000
8126000000
807000000
000010000
0008126000
000807000
000000070
0000001884
00000018011
,
000100000
000010000
000001000
000000100
000000010
000000001
700000000
070000000
007000000

G:=sub<GL(9,GF(19))| [1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[1,0,8,0,0,0,0,0,0,0,11,11,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,18,7,0,0,0,0,0,0,11,8,1,0,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,0,18,11,11,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,4,0,1],[0,8,8,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,6,7,0,0,0,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,6,7,0,0,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,0,7,8,0,0,0,0,0,0,0,0,4,11],[0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0] >;

C32.5He3 in GAP, Magma, Sage, TeX

C_3^2._5{\rm He}_3
% in TeX

G:=Group("C3^2.5He3");
// GroupNames label

G:=SmallGroup(243,29);
// by ID

G=gap.SmallGroup(243,29);
# by ID

G:=PCGroup([5,-3,3,-3,-3,-3,405,121,186,542,457,282,2163]);
// Polycyclic

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

Export

Subgroup lattice of C32.5He3 in TeX
Character table of C32.5He3 in TeX

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