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Cayley diagram

(C3⋊GL(2,𝔽3))⋊C3 #88601

order432
isoclinismisoclinic class of (C3⋊#226)⋊C3
automorphismAut(G) = C2×((C3⋊S4)⋊C3)⋊C2 (order 864) · Inn(G) = (C3⋊S4)⋊C3 · Out(G) = C22
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect product(C3⋊GL(2,𝔽3))⋊C3

Splittings

FittingC32×Q8 ⋊ D3
derivedC3×SL(2,𝔽3) ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2(C3⋊S4)⋊C3
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C2×C3
Frattini, socleC2×C3C3×S4
FittingC32×Q8D3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,8,3,4,7,6,5)(10,15,17,12,13,16)(11,14);(2,5,8)(3,6,7)(9,10,12)(11,13,16)(14,17,15)

Polycyclic presentation:

2,3^3,2^3|e^2=g,f^2=g,c^a=c^2,d^a=d^2,e^a=f,f^a=e,c^b=cd,e^c=fg,f^c=ef,f^e=fg

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11722255, #11722257, #11722258, …
maximal normal subgroup of#11722255, #11722257, #11722258, …
maximal quotient of(C3⋊#859)⋊C3, C2×(C3⋊GL(2,𝔽3))⋊C3, #13046956, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 248);