#72213328

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₃² ⋊ C₂×C₃⋊D₃
derived	(C₂²⋊A₄)⋊H(3,𝔽₃) ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	(C₂²⋊A₄)⋊H(3,𝔽₃)	C₂²
Frattini	C₃	D₃×A₄⋊S₄
Fitting	C₂⁴×C₃²	C₂×C₃⋊D₃
socle	C₂⁴×C₃	D₃×C₃⋊D₃
LCS	G ▷ (C₂²⋊A₄)⋊H(3,𝔽₃)
UCS	C₁

1419 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 1419 conjugacy classes of subgroups — too many to draw. Only its 42 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:

2^2,3^4,2^4|c^b=c^2,d^a=d^2,d^b=d^2,e^b=e^2,e^d=ef,f^a=f^2,g^c=h,g^e=h,h^b=gh,h^c=gh,h^e=gh,i^b=hij,i^c=j,i^e=ghi,j^b=gj,j^c=ij,j^e=gj

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72213328
maximal normal subgroup of	C₂×#72213328
maximal quotient of	#72220377, C₂×#72213328, #72222920
holomorph of	C₃×A₄

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