#72216768

order	2640
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₈×C₁₁ ⋊ C₂×C₅
Hall	#10587194 ⋊ C₅
Hall	C₃×C₁₁⋊C₅ ⋊ M₁₆
Hall	C₁₁⋊C₅ ⋊ ΓL(1,𝔽₂₅)
Hall	C₃×C₁₁ ⋊ C₅×M₁₆
Sylow	C₁₁ ⋊ C₅×ΓL(1,𝔽₂₅)
Sylow	C₃ ⋊ #4313⋊C₅

Subgroups

	subgroup	G/H
center, Frattini	C₄	C₂×C₃⋊AGL(1,𝔽₁₁)
commutator, socle	C₂×C₃×C₁₁	C₂×C₄×C₅
Fitting	C₃×C₈×C₁₁	C₂×C₅
prime radical	C₂×C₃⋊AGL(1,𝔽₁₁)	C₄
cosocle	C₃×C₄×C₁₁	C₂²×C₅
hypercenter	C₈	C₃⋊AGL(1,𝔽₁₁)
hypocenter	C₃×C₁₁	C₅×M₁₆
LCS	G ▷ C₂×C₃×C₁₁ ▷ C₃×C₁₁
UCS	C₁ ◁ C₄ ◁ C₈

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,5,2^2,3,11|a^2=d,d^2=e,b^a=be,f^b=f^2,g^b=g^10,g^c=g^9

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	M₁₆×C₃⋊AGL(1,𝔽₁₁), D₃×#4313⋊C₅, ΓL(1,𝔽₂₅)×AGL(1,𝔽₁₁), …
maximal normal subgroup of	M₁₆×C₃⋊AGL(1,𝔽₁₁), D₃×#4313⋊C₅, ΓL(1,𝔽₂₅)×AGL(1,𝔽₁₁), …

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");;
gap> coll := SingleCollector(F, [2, 2, 5, 2, 2, 3, 11]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7^10);;
gap> SetConjugate(coll, 7, 3, F.7^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);