order	224
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) = C₂³×C₃×#120 (order 768) · Inn(G) = C₂³ · Out(G) = C₂⁵×C₃
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₇×C₂²⋊Q₈
unambiguous semidirect product	C₂²⋊(C₇×Q₈)

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₇
prime radical	C₂³×C₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

74 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 157);