C₁₁×#2328 #13065166

order	1408
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₁₁ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₁₁×#2328

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₁
Frattini, cosocle	C₂³	C₂⁴×C₁₁
prime radical	C₂×C₁₁×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ G

740 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 296 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) = 704

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

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

Conjugacy classes & character table

484 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14649279, #14649299, #14650472, …
maximal subgroup of	C₁₁×#9101, C₁₁×#9105, C₁₁×#9295, …
maximal normal subgroup of	C₁₁×#9101, C₁₁×#9105, C₁₁×#9295, …
maximal quotient of	C₁₁×#13779, C₁₁×#13785, C₁₁×#13791, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1408, 1060);