#72210357

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₃² ⋊ C₂²×D₃
derived	C₃²×C₂²⋊A₄ ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×C₂²⋊A₄	C₂³
Frattini	C₂²	D₃²×S₄
Fitting	C₂⁴×C₃²	C₂²×D₃
socle	C₂²×C₃²	C₂²×S₄
LCS	G ▷ C₃²×C₂²⋊A₄
UCS	C₁

2188 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 2188 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3^3,2^4|d^a=d^2,d^c=d^2,e^a=d^2e,e^c=e^2,f^b=f^2,g^a=gij,g^b=gh,g^f=h,h^a=hi,h^f=gh,i^f=j,j^b=ij,j^f=ij

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72210357, #72220866, #72220940, …
maximal normal subgroup of	C₃×#72210357, D₃²×#1020⋊C₂, #72241545, …
maximal quotient of	C₃×#72210357, #72220866, #72220940, …
holomorph of	C₂²×D₃

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