order	504
isoclinism	isoclinic class of (C₃⋊D₃)⋊D₇
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

	subgroup	G/H
center, hypercenter	C₂	(C₃⋊D₃)⋊D₇
commutator, cosocle, hypocenter	C₃²×C₇	C₂³
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇	C₂²
LCS	G ▷ C₃²×C₇
UCS	C₁ ◁ C₂

138 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 3 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 193);