#72247288

order	14520
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Hall	#72247167 ⋊ C₅
Hall	C₃×#10597243 ⋊ D₄
Hall	#10597243 ⋊ #67
Hall	C₃×C₁₁² ⋊ C₅×D₄
Sylow	C₁₁² ⋊ C₅×#67
Sylow	C₃ ⋊ #72213666

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#11925895⋊D₃
commutator, Fitting, socle, cosocle	C₂×C₃×C₁₁²	C₂²×C₅
hypocenter	C₃×C₁₁²	C₅×D₄
LCS	G ▷ C₂×C₃×C₁₁² ▷ C₃×C₁₁²
UCS	C₁ ◁ C₂

228 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 132

Polycyclic presentation:

2^2,5,2,3,11^2|b^a=bd,e^b=e^2,f^a=f^10,f^b=f^10,f^c=f^3,g^a=f^2g,g^b=g^10,g^c=g^3

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72247294, #72247317, D₃×#72213666
maximal normal subgroup of	#72247294, #72247317, D₃×#72213666

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