C₇×#9991 #13187207

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#9991

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁴×C₇	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

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

normal subgroups only

This group has 698 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,17,22,27)(18,28,32,39)(19,33,37,42)(20,26,21,25)(23,31,40,30)(24,35,43,36)(29,38,45,46)(34,47,44,41);(12,15)(16,18,22,32)(17,23,27,40)(19,34,37,44)(20,31,21,30)(24,41,43,47)(25,39,26,28)(29,36,45,35)(33,46,42,38);(8,10)(9,12)(11,14)(13,15)(16,19,22,37)(17,24,27,43)(18,29,32,45)(20,35,21,36)(23,38,40,46)(25,42,26,33)(28,41,39,47)(30,34,31,44)

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, 2979);