C₂²×D₂₁ #4286

order	168
isoclinism	isoclinic class of D₂₁
automorphism	Aut(G) = D₃×S₄×AGL(1,𝔽₇) (order 6048) · Inn(G) = D₂₁ · Out(G) = C₂×C₃×S₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

C₂²×D₂₁

	subgroup	G/H
center, hypercenter	C₂²	D₂₁
commutator, cosocle, hypocenter	C₃×C₇	C₂³
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₇	C₂
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂²

64 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 3 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 56);