C₃⋊GL(2,𝔽₃) #3782

order	144
isoclinism	isoclinic class of C₃⋊#226
automorphism	Aut(G) = C₂×((C₃⋊S₄)⋊C₃)⋊C₂ (order 864) · Inn(G) = C₃⋊S₄ · Out(G) = C₂×D₃
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

Fitting	C₃×Q₈ ⋊ D₃
derived	C₃×SL(2,𝔽₃) ⋊ C₂

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊S₄
commutator, cosocle, hypocenter	C₃×SL(2,𝔽₃)	C₂
Fitting	C₃×Q₈	D₃
socle	C₂×C₃	S₄
LCS	G ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

46 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 3 generators on 11 points:

(1,2,8,3,4,7,6,5)(10,11);(9,10,11);(2,5,8)(3,6,7)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 125);