#72269511

order	3520
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₁₁×C₂²⋊C₄ ⋊ C₂×C₅
Hall	#10623021 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ #546
Sylow	C₁₁ ⋊ C₅×#546

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₁₁)
commutator, socle, cosocle	C₂²×C₁₁	C₂⁴×C₅
Fitting	C₂×C₁₁×C₂²⋊C₄	C₂×C₅
hypercenter	C₂×C₂²⋊C₄	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#546
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 176

A minimal faithful permutation representation — 4 generators on 23 points:

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

Polycyclic presentation:

2^4,5,2^2,11|c^2=g,d^2=g,c^b=cg,d^a=df,d^c=dg,h^c=h^10,h^e=h^9

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72269537, #546×AGL(1,𝔽₁₁), #72269586
maximal normal subgroup of	#72269537, #546×AGL(1,𝔽₁₁), #72269586

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 5, 2, 2, 11]);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 8, 3, F.8^10);;
gap> SetConjugate(coll, 8, 5, F.8^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);