Finite Group Databasebeta

← Prev (#66347) Next (#66349) →

Cayley diagram

#4176⋊C2 #66348

order324
isoclinismisoclinic class of #4176⋊C2
automorphismAut(G) = (#4176⋊C2)⋊C3 (order 972) · Inn(G) = G · Out(G) = C3
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#4176⋊C2
unambiguous semidirect product#4177⋊C2
unambiguous semidirect product#4184⋊C2

Splittings

Fitting, derived, Sylow#731 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#731C22
FrattiniC32D32
socleC3((C3⋊D3)⋊C3)⋊C2
LCSG ▷ #731
UCSC1

54 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:

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

Polycyclic presentation:

2^2,3^4|c^3=f^2,d^a=d^2,e^a=e^2f,c^b=c^2f,e^b=e^2,f^b=f^2,d^c=de,e^d=ef

Conjugacy classes & character table

17 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#4176⋊C2, C3×#4176⋊C2, (#4176⋊C2)⋊C3, …
maximal normal subgroup ofC2×#4176⋊C2, C3×#4176⋊C2, (#4176⋊C2)⋊C3, …
maximal quotient of#4176⋊C4, #4184⋊C4, #4177⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 40);