C₇×#9976 #13187192

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9974
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#9976

Subgroups

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

716 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 102 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)(16,17,22,26)(18,27,31,39)(19,32,36,42)(20,37,21,25)(23,30,40,29)(24,34,43,35)(28,45,46,38)(33,44,47,41);(12,15)(16,18)(17,23)(19,33)(20,30)(21,29)(22,31)(24,41)(25,39)(26,40)(27,37)(28,34)(32,38)(35,46)(36,47)(42,45)(43,44);(8,10)(9,12)(11,14)(13,15)(16,19,22,36)(17,24,26,43)(18,28,31,46)(20,34,21,35)(23,38,40,45)(25,42,37,32)(27,44,39,41)(29,33,30,47)

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, 2964);