C₇×#547 #90568

order	448
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

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

292 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 224

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1330);