order	504
isoclinism	isoclinic class of (C₂²⋊C₉)⋊C₂
automorphism	Aut(G) = C₂×C₃×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 1296) · Inn(G) = (C₂²⋊C₉)⋊C₂ · Out(G) = C₂×C₃²
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×(C₂²⋊C₉)⋊C₂
unambiguous semidirect product	(C₂²⋊C₉)⋊(C₂×C₇)

Splittings

	subgroup	G/H
center, hypercenter	C₇	(C₂²⋊C₉)⋊C₂
commutator, cosocle, hypocenter	C₂²⋊C₉	C₂×C₇
Frattini	C₃	C₇×S₄
Fitting, socle	C₂²×C₃×C₇	D₃
LCS	G ▷ C₂²⋊C₉
UCS	C₁ ◁ C₇

36 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 126

A minimal faithful permutation representation — 2 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 54);