#72220659

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	A₄×C₂²⋊H(3,𝔽₃) ⋊ C₂²

Subgroups

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

1462 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 1462 conjugacy classes of subgroups — too many to draw. Only its 43 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^a=c^2,c^b=c^2,d^a=d^2,e^a=e^2,e^c=ef,f^b=f^2,g^d=h,g^e=gh,h^a=gh,h^d=gh,h^e=g,i^a=gi,i^d=gi,i^e=j,j^a=hij,j^d=ghj,j^e=ij

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×((C₃⋊S₄)⋊C₃)⋊C₂
maximal normal subgroup of	S₄×((C₃⋊S₄)⋊C₃)⋊C₂
maximal quotient of	#72222920

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