C₁₁×#2005 #13064843

order	1408
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₁×#2005

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₁	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₁
prime radical	C₂×C₁₁×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₂²×C₄×C₁₁ ◁ G

362 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 352

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

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

Conjugacy classes & character table

286 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₁₁×#2005, #14780683, #14780699, …
maximal subgroup of	C₂×C₁₁×#2005, C₁₁×#23399, C₁₁×#23401, …
maximal normal subgroup of	C₂×C₁₁×#2005, C₁₁×#23399, C₁₁×#23401, …
maximal quotient of	C₁₁×#9917, C₁₁×#9919, C₁₁×#9925, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1408, 737);