C₇×#126 #6529

order	224
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) = C₂³×C₃×C₂²⋊A₄ (order 1152) · Inn(G) = C₂³ · Out(G) = C₂²×C₃×A₄
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

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

60 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 161);