C₇×#8535 #13185751

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₇×#8535

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#379
commutator	C₂⁴	C₂²×C₄×C₇
Frattini, cosocle	C₂⁵	C₂³×C₇
prime radical	C₇×#3490	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂⁵×C₇ ◁ G

1062 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 224

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

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1523);