C₇×#131 #6534

order	224
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂³×C₃×S₄ (order 576) · Inn(G) = C₂² · Out(G) = C₂³×C₃×D₃
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

62 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,19,20,21,22,23);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16);(3,9)(6,12)(10,15)(13,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 166);