C₁₁×#7186 #14584379

order	2816
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₁×#7186

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₁₁	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₁₁
Frattini, cosocle	C₂³×C₈	C₂²×C₁₁
socle	C₂³×C₁₁	C₂²⋊C₈
prime radical	C₂⁴×C₁₁	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁ ◁ C₂³×C₈×C₁₁ ◁ G

326 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 55
transitive degree	μ_t(G) = 1408

A minimal faithful permutation representation — 2 generators on 55 points:

(1,2,3,4,5,6,7,8,9,10,11)(16,17)(18,22,23,20)(19,21)(24,25,28,33,29,34,43,51,30,35,44,52,27,32,41,49)(26,36,38,53,37,47,54,50,39,48,42,55,40,31,45,46);(12,13,14,15)(16,18)(17,20)(19,23)(21,22)(24,26)(25,31)(27,40)(28,42)(29,37)(30,39)(32,48)(33,50)(34,36)(35,47)(38,44)(41,54)(43,45)(46,52)(49,53)(51,55)

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, 174);