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Cayley diagram

#4184⋊C3 #92332

order486
isoclinismisoclinic class of #4184⋊C3
automorphismAut(G) = (#4176⋊C2)⋊C3 (order 972) · Inn(G) = G · Out(G) = C2
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#4184⋊C3

Splittings

Fitting, Sylow#6910 ⋊ C2
derived#731 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#731C2×C3
FrattiniC32C3×C3⋊D3
Fitting#6910C2
socleC3C3×#272
LCSG ▷ #731
UCSC1

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 27 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(#4176⋊C2)⋊C3, C2×#4184⋊C3, #13091114, …
maximal normal subgroup of(#4176⋊C2)⋊C3, C2×#4184⋊C3, C3×#4184⋊C3, …
maximal quotient of#66331⋊C3, C2×#4184⋊C3, #10624590⋊C2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 177);