C₇⋊(C₄²⋊C₃) #66588

order	336
isoclinism	isoclinic class of C₇⋊(C₄²⋊C₃)
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₄

Aliases

unambiguous semidirect product	C₇⋊(C₄²⋊C₃)
unambiguous semidirect product	C₄²⋊ΓL(1,𝔽₈)

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄²×C₇	C₃
Frattini	C₂²	C₂²⋊ΓL(1,𝔽₈)
socle	C₂²×C₇	A₄
LCS	G ▷ C₄²×C₇
UCS	C₁

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 2 generators on 19 points:

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

Polycyclic presentation:

3,2^2,7,2^2|b^2=ef,c^2=e,b^a=c,c^a=bcf,d^a=d^2,e^a=f,f^a=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 57);