C₇×#10667 #13187883

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#10667

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#399
prime radical	C₇×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 354 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) = 47
transitive degree	μ_t(G) = 448

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

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

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, 3655);