C₂×C₇×C₄⋊C₄ #6519

order	224
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×C₃×#347042 (order 3072) · Inn(G) = C₂² · Out(G) = C₂²×C₃×B(4,𝔽₂)
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

C₂×C₇×C₄⋊C₄

92 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 151);