C₃²×D₇ #1262

order	126
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = AGL(1,𝔽₇)×GL(2,𝔽₃) (order 2016) · Inn(G) = D₇ · Out(G) = C₃×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

C₃²×D₇

Fitting, socle, Hall	C₃²×C₇ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂×C₃²

	subgroup	G/H
center, hypercenter	C₃²	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₃²

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 63

A minimal faithful permutation representation — 2 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(126, 11);