C₇×#13309 #13190525

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13309

Subgroups

	subgroup	G/H
center	C₄×C₇	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1999
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

568 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 114 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)(10,18)(11,22)(12,27)(13,16)(14,19)(15,25)(17,24)(20,34)(21,36)(23,33)(26,37)(28,31)(29,35)(30,38)(32,39);(8,10)(9,14)(11,23)(12,28)(13,20)(15,31)(16,29)(17,33)(18,22)(19,25)(21,24)(26,38)(27,30)(32,36)(34,37)(35,39);(14,29)(18,34)(30,33)(31,36)

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6297);