C₇×#397 #90043

order	448
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

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

170 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

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

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 805);