C₂×C₇×#123 #90545

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, socle	C₂³×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

342 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

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

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1307);