C₇×#10393 #13187609

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10393

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#393
commutator, Frattini, cosocle	C₂⁵	C₂³×C₇
prime radical	C₂⁵×C₇	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂⁵×C₇ ◁ G

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

normal subgroups only

This group has 526 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) = 39
transitive degree	μ_t(G) = 224

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

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3381);