C₇×#489 #90135

order	448
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₄

Aliases

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₂×C₇×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

100 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 897);