C₇×#10301 #13187517

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10300
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10301

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

554 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 23
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3,4,5,6,7)(8,9,12,10)(11,13,15,14)(16,17)(18,21)(19,22)(20,23);(9,13)(10,14)(11,15)(16,17)(18,22)(19,21)(20,23);(8,10)(9,12)(11,13)(14,15)(17,21)(18,19)(22,23)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3289);