#72247308

order	29040
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×C₁₁² ⋊ C₂²×C₅
Hall	#72247187 ⋊ C₅
Hall	C₃×#10597243 ⋊ C₂×D₄
Hall	#10597243 ⋊ C₂×#67
Hall	C₃×C₁₁² ⋊ C₂×C₅×D₄
Sylow	C₁₁² ⋊ C₂×C₅×#67
Sylow	C₃ ⋊ #72213693

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#20300008
commutator, socle, cosocle	C₂×C₃×C₁₁²	C₂³×C₅
Fitting	C₂²×C₃×C₁₁²	C₂²×C₅
hypercenter	C₂²	#20300008
hypocenter	C₃×C₁₁²	C₂×C₅×D₄
LCS	G ▷ C₂×C₃×C₁₁² ▷ C₃×C₁₁²
UCS	C₁ ◁ C₂ ◁ C₂²

752 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72213693
maximal normal subgroup of	D₃×#72213693

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, 5, 2, 3, 11, 11]);;
gap> SetPower(coll, 1, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> SetConjugate(coll, 7, 3, F.7^10);;
gap> SetConjugate(coll, 7, 4, F.7^3);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^10);;
gap> SetConjugate(coll, 8, 4, F.8^3);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);