C₂²×C₁₁×H(3,𝔽₄) #14639881

order	2816
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₁₁ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂²×C₁₁×H(3,𝔽₄)

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₁₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁ ◁ G

10246 conjugacy classes of subgroups; only the 5774 normal subgroups are shown.

normal subgroups only

This group has 10246 conjugacy classes of subgroups — too many to draw. Only its 5774 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) = 704

A minimal faithful permutation representation — 6 generators on 31 points:

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

Conjugacy classes & character table

836 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2816, 55676);