C₇⋊#622 #92665

order	504
isoclinism	isoclinic class of C₇⋊#157
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₃

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₇⋊#157
commutator, hypocenter	C₃²×C₇	C₈
Fitting, socle, prime radical	C₂×C₃²×C₇	C₄
cosocle	C₇×C₃⋊Dic₃	C₂
LCS	G ▷ C₃²×C₇
UCS	C₁ ◁ C₂

28 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 2 generators on 21 points:

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

Polycyclic presentation:

2^3,3^2,7|a^2=b,b^2=c,d^a=d^2e,e^a=de,f^a=f^6,d^b=d^2,e^b=e^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 91);