C₇×#7824 #13185040

order	1792
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7824

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂²×C₇×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

418 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 896

A minimal faithful permutation representation — 3 generators on 39 points:

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

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 812);