C₇×#7834 #13185050

order	1792
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#7834

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

396 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 94 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)(9,12)(10,14)(13,15)(17,22)(18,23)(19,20)(24,25,29,32)(26,33,35,38)(27,31,28,36)(30,37,39,34);(10,14)(12,15)(16,17,21,18)(19,23,20,22)(24,26)(25,30)(27,34)(28,37)(29,35)(31,38)(32,39)(33,36);(8,9)(10,15)(11,13)(12,14)(16,18,21,17)(19,23,20,22)(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, 822);