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Cayley diagram

#3458⋊C3 #85650

order384
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22

Aliases

unambiguous semidirect product#3458⋊C3

Splittings

Fitting, Sylow#3458 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2×C4C22⋊A4
commutator, cosocle, hypocenter#543C2×C3
Frattini, socleC22C2×C22⋊A4
Fitting#3458C3
prime radical#543⋊C3C2
LCSG ▷ #543
UCSC1 ◁ C2×C4

216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2,3,2^6|a^2=g,c^2=h,e^2=gh,c^b=cdefgh,d^b=fh,e^b=cd,f^b=dfh,d^c=dg,e^c=eh,f^c=fg,e^d=eg

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11391⋊C3, #11395⋊C3, #23467⋊C3, …
maximal normal subgroup of#11391⋊C3, #11395⋊C3, #23467⋊C3, …
maximal quotient ofC4×#543⋊C3, #36674⋊C3, #52287⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18226);