C₂²×AGL(1,𝔽₈) #6563

order	224
isoclinism	isoclinic class of AGL(1,𝔽₈)
automorphism	Aut(G) = D₃×AΓL(1,𝔽₈) (order 1008) · Inn(G) = AGL(1,𝔽₈) · Out(G) = C₃×D₃
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂²×AGL(1,𝔽₈)
unambiguous semidirect product	C₂⁵⋊C₇

Fitting, socle, Sylow	C₂⁵ ⋊ C₇
derived	C₂³ ⋊ C₂²×C₇

	subgroup	G/H
center, hypercenter	C₂²	AGL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂³	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₂⁵	C₇
LCS	G ▷ C₂³
UCS	C₁ ◁ C₂²

72 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 28

A minimal faithful permutation representation — 2 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 195);