C₇×#7828 #13185044

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

Aliases

direct product

C₇×#7828

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

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

normal subgroups only

This group has 404 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)(17,22)(18,23)(19,20)(24,25,28,31)(26,29,33,36)(27,30,34,37)(32,38,39,35);(16,17,21,18)(19,23,20,22)(24,26,28,33)(25,29,31,36)(27,32,34,39)(30,35,37,38);(9,12)(10,15)(13,14)(16,18,21,17)(19,23,20,22)(24,27)(25,30)(26,32)(28,34)(29,35)(31,37)(33,39)(36,38)

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, 816);