C₇×#533 #90554

order	448
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

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

222 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1316);