C₁₁×#33567 #14610760

order	2816
nilpotency class	4
isoclinism	isoclinic class of #2204
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₁₁×#33567

Subgroups

	subgroup	G/H
center	C₄×C₁₁	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂⁴×C₁₁
socle	C₂×C₁₁	C₂×B(4,𝔽₂)
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₁ ◁ C₂×C₄×C₁₁ ◁ C₂×C₁₁×#41 ◁ G

1184 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 27
transitive degree	μ_t(G) = 176

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

440 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2816, 26555);