C₇×#12460 #13189676

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12433
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#12460

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×C₈×Q₈ ◁ G

200 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 1792

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

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

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, 5448);