order	224
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	Aut(G) = C₂×C₃×#10596770⋊C₂ (order 6912) · Inn(G) = C₂⁴ · Out(G) = C₂×C₃×GO⁺(4,𝔽₂)
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁴×C₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ G

166 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 4 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 193);