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Cayley diagram

#4207⋊C2 #66433

order324
isoclinismisoclinic class of #4207⋊C2
automorphismAut(G) = (#66419⋊C2)⋊C3 (order 1944) · Inn(G) = G · Out(G) = C2×C3
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#4207⋊C2
unambiguous semidirect product#4208⋊C2

Splittings

Fitting, derived, Sylow#737 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#737C22
Frattini, socleC3#1113
LCSG ▷ #737
UCSC1

71 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

21 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#66419⋊C2, C2×#4207⋊C2, #92181⋊C2, …
maximal normal subgroup of#66419⋊C2, C2×#4207⋊C2, C3×#4207⋊C2, …
maximal quotient of#10619905, #4207⋊C4, #10619907, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 125);