C₇×#10384 #13187600

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#10384

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

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

normal subgroups only

This group has 438 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) = 448

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)(16,17,19,22)(18,24)(20,23,28,30)(21,29)(25,26)(27,31);(10,15)(12,13)(16,17)(18,25,27,21)(19,22)(20,23)(24,26,31,29)(28,30);(9,12)(10,15)(13,14)(16,18)(17,21)(19,27)(20,29)(22,25)(23,31)(24,30)(26,28)

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, 3372);