C₇×#10666 #13187882

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

Aliases

direct product

C₇×#10666

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

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, 3654);