C₇×#10339 #13187555

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10336
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10339

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂⁵	C₂³×C₇
prime radical	C₂⁵×C₇	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂⁵×C₇ ◁ G

738 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7)(8,9,12,17)(10,14,21,28)(11,15,22,30)(13,18,24,16)(19,33,34,29)(20,26,36,27)(23,32,37,31)(25,38,39,35);(8,10,12,21)(9,14,17,28)(11,19,22,34)(13,20,24,36)(15,29,30,33)(16,27,18,26)(23,35,37,38)(25,32,39,31);(8,11,13,23)(9,15,18,32)(10,19,20,35)(12,22,24,37)(14,25,26,33)(16,31,17,30)(21,34,36,38)(27,29,28,39)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3327);