C₇×#127 #6530

order	224
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) = C₂×C₃×AGL(2,ℤ/4) (order 9216) · Inn(G) = C₂³ · Out(G) = C₂²×C₃×#1020
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

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

108 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 162);