order	224
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×C₃×#41 (order 192) · Inn(G) = C₂² · Out(G) = C₂⁴×C₃
homology	Abelianization (H₁) = C₂×C₇×C₈ · Schur multiplier (H₂) = C₁

Aliases

C₇×M₃₂

Subgroups

26 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 59);