C₇×#108 #6425

order	224
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂²×C₃×AGL(1,ℤ/8) (order 384) · Inn(G) = D₄ · Out(G) = C₂⁴×C₃
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

30 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 57);