C₇×#532 #90553

order	448
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center	C₂×C₄×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
socle	C₂²×C₇	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

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

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 170 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1315);