C₇×#10228 #13187444

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂³×C₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

828 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.

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

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3,4,5,6,7)(12,13)(14,16)(15,18)(17,19)(20,21)(22,24)(23,26)(25,27);(8,9)(10,11)(13,16)(14,19)(17,18);(9,11)(21,24)(22,27)(25,26)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3216);