C₂×C₇×#41 #6560

order	224
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×C₃×#11713200 (order 4608) · Inn(G) = C₂² · Out(G) = C₂×C₃×D₄×S₄
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂×C₇×#41

	subgroup	G/H
center	C₂×C₄×C₇	C₂²
commutator, Frattini, cosocle	C₂	C₂⁴×C₇
socle	C₂²×C₇	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

164 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 4 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 192);