C₇×#10535 #13187751

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10530
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#10535

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

342 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 896

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

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

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, 3523);