C₇⋊GL(2,𝔽₃) #66650

order	336
isoclinism	isoclinic class of C₇⋊#226
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₇⋊S₄
commutator, cosocle, hypocenter	C₇×SL(2,𝔽₃)	C₂
Fitting	C₇×Q₈	D₃
socle	C₂×C₇	S₄
LCS	G ▷ C₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

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

Polycyclic presentation:

2,3,7,2^3|d^2=f,e^2=f,b^a=b^2,c^a=c^6,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(336, 119);