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Cayley diagram

#6913⋊C2 #92287

order486
automorphismAut(G) = C32×(C3⋊D3)⋊C3 (order 486) · Inn(G) = C3×(C3⋊D3)⋊C3 · Out(G) = C3
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect product#6913⋊C2

Splittings

Fitting, Sylow#6913 ⋊ C2

Subgroups

subgroupG/H
center, socleC3C3×(C3⋊D3)⋊C3
commutator, cosocle, hypocenterH(3,𝔽3)C2×C32
FrattiniC32C32×D3
Fitting#6913C2
hypercenterC9(C3⋊D3)⋊C3
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C3 ◁ C9

69 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 27

A minimal faithful permutation representation — 2 generators on 27 points:

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

Polycyclic presentation:

2,3^5|c^3=f,d^a=d^2,e^a=e^2f,c^b=cf,d^b=de,e^d=ef

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6913⋊C2, #13091321, C3×#6913⋊C2, …
maximal normal subgroup ofC2×#6913⋊C2, C3×#6913⋊C2, #13091641, …
maximal quotient of#6913⋊C4, C2×#6913⋊C2, #10624622⋊C2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 132);