C₇×#10044 #13187260

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10044

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 466 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)(10,16)(11,19)(12,15)(13,18)(14,21)(17,23)(20,22)(24,25,29,33)(26,34,37,39)(27,31,28,32)(30,36,38,35);(13,22)(16,23)(24,26,29,37)(25,30,33,38)(27,36,28,35)(31,34,32,39);(8,10)(9,13)(11,20)(12,17)(14,23)(15,22)(16,19)(18,21)(25,31)(26,35)(27,28)(32,33)(34,39)(36,37)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3032);