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Cayley diagram

#4178⋊C2 #66349

order324
isoclinismisoclinic class of #4178⋊C2
automorphismAut(G) = (#4178⋊C2)⋊C3 (order 972) · Inn(G) = G · Out(G) = C3
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#4178⋊C2
unambiguous semidirect product#4179⋊C2
unambiguous semidirect product#4185⋊C2

Splittings

Fitting, derived, Sylow#732 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#732C22
FrattiniC32D32
socleC3((C3⋊D3)⋊C3)⋊C2
LCSG ▷ #732
UCSC1

56 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)(4,13,23,5,14,22)(7,20,25,9,17,27)(10,19,21,16,12,26);(1,10,6,5,9,7,4,2,16)(3,27,19,22,8,24,14,21,17,11,20,26,13,18,15,23,12,25)

Polycyclic presentation:

2^2,3^4|c^3=f,d^a=d^2,e^a=e^2f,c^b=c^2f^2,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×#4178⋊C2, #92191⋊C2, C3×#4178⋊C2, …
maximal normal subgroup ofC2×#4178⋊C2, C3×#4178⋊C2, (#4178⋊C2)⋊C3, …
maximal quotient of#4178⋊C4, #4185⋊C4, #4179⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 41);