C₇×#11390 #13188606

order	1792
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₇×#11390

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	#120
commutator	C₂⁴	C₂²×C₄×C₇
Frattini, cosocle	C₂⁵	C₂³×C₇
prime radical	C₂⁵×C₇	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂⁵×C₇ ◁ G

1358 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 1358 conjugacy classes of subgroups — too many to draw. Only its 270 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) = 448

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

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

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4378);