C₇×#394 #90040

order	448
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

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

238 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 802);