C₇×#10686 #13187902

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#10686

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

410 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 162 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)(8,9,11,13)(10,14,15,12)(24,25,29,30,28,33,27,32)(26,34,37,31,36,39,35,38);(8,10,11,15)(9,12,13,14)(16,17,19,22)(18,20,23,21)(25,30)(27,29)(31,34)(32,33)(35,37)(38,39);(17,20)(18,23)(21,22)(24,26,28,36)(25,31,33,38)(27,37,29,35)(30,34,32,39)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3674);