C₁₁×#2274 #13065112

order	1408
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₁₁ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₁₁×#2274

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₁	C₂⁴
commutator	C₂	C₂²×C₄²×C₁₁
Frattini, cosocle	C₂³	C₂⁴×C₁₁
prime radical	C₂⁴×C₁₁	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ G

696 conjugacy classes of subgroups; only the 520 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 520 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)(16,17,19,18)(20,21,24,22)(23,26,27,25);(12,13,14,15)(16,18,19,17)(21,22)(25,26);(20,22,24,21)(23,26,27,25);(20,23)(21,25)(22,26)(24,27)

Conjugacy classes & character table

748 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14647235, #14651213, #14651223, …
maximal subgroup of	C₁₁×#8735, C₁₁×#9408, C₁₁×#9410, …
maximal normal subgroup of	C₁₁×#8735, C₁₁×#9408, C₁₁×#9410, …
maximal quotient of	C₁₁×#13749, C₁₁×#13750, C₁₁×#13776, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1408, 1006);