C₇×#535 #90556

order	448
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

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

334 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 4 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1318);