C₇×#10382 #13187598

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#10382

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂×C₇×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

446 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 31
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,3,4,5,6,7)(8,9)(10,12)(11,14)(13,15)(16,17)(18,24,26,30)(19,22)(20,23)(21,27,25,28)(29,31);(10,15)(12,13)(16,17)(18,25,26,21)(19,22)(20,23)(24,27,30,28)(29,31);(9,12)(10,15)(13,14)(16,18)(17,21)(19,28)(20,26)(22,30)(23,25)(24,31)(27,29)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3370);