C₇×#8543 #13185759

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₇×#8543

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#379
commutator	C₂⁴	C₂²×C₄×C₇
Frattini, cosocle	C₂⁵	C₂³×C₇
prime radical	C₂³×C₇×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂⁵×C₇ ◁ G

1266 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 31
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 31 points:

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

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, 1531);