#72276137

order	11664
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×#10624749 ⋊ C₂²
Sylow	#10624749 ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂×#72210336
commutator, cosocle	C₂×#10624749	C₂³
Frattini	C₂×C₃³	C₂×D₃×C₃⋊D₃
Fitting	C₂²×#10624749	C₂²
socle	C₂×C₃	C₂×#11757730
hypercenter	C₂²	#72210336
hypocenter	#10624749	C₂×D₄
LCS	G ▷ C₂×#10624749 ▷ #10624749
UCS	C₁ ◁ C₂ ◁ C₂²

2056 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 2056 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 108

Polycyclic presentation:

2^4,3^6|a^2=d,g^3=j,b^a=bd,e^a=e^2,f^a=f^2,f^e=fi,g^a=gh,g^c=g^2j^2,g^e=gj,g^f=gh,h^a=h^2,h^c=h^2,h^e=hj,i^g=ij^2,j^c=j^2

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#72210336
maximal normal subgroup of	D₄×#72210336

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", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7^2*F.10^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10^2);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);