C₇⋊#226 #66649

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

Aliases

C₇ ⋊ #226

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

26 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 2 generators on 23 points:

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

Polycyclic presentation:

2,3,7,2^3|a^2=f,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, 118);