C₇×#13331 #13190547

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13331

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

650 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 224

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

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6319);