C₁₁⋊C₃₂ #66859

order	352
isoclinism	isoclinic class of D₁₁
automorphism	Aut(G) = C₂×C₈×AGL(1,𝔽₁₁) (order 1760) · Inn(G) = D₁₁ · Out(G) = C₂×C₅×C₈
homology	Abelianization (H₁) = C₃₂ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₁₁⋊C₃₂

Splittings

derived, Sylow

C₁₁ ⋊ C₃₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁₆	D₁₁
commutator, hypocenter	C₁₁	C₃₂
Fitting, cosocle	C₁₁×C₁₆	C₂
socle, prime radical	C₂×C₁₁	C₁₆
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₁₆

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 352

A minimal faithful permutation representation — 2 generators on 43 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,14,18,15,19,22,28,16,20,23,29,25,31,34,38,17,21,24,30,26,32,35,39,27,33,36,40,37,41,42,43);(1,2,4,6,8,10,11,9,7,5,3)

Polycyclic presentation:

2^5,11|a^2=b,b^2=c,c^2=d,d^2=e,f^a=f^10

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₁×C₃₂, #10622085, C₂×C₁₁⋊C₃₂, …
maximal normal subgroup of	D₁₁×C₃₂, #10622085, C₂×C₁₁⋊C₃₂, …
maximal quotient of	C₁₁⋊C₆₄, C₂×C₁₁⋊C₃₂, C₃×C₁₁⋊C₃₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 1);

C11⋊C32 #66859