#72220447

order	6912
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting

C₃²×C₂²⋊C₄ ⋊ GL(2,𝔽₃)

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(2,𝔽₃)
commutator, cosocle	C₂²×PSU(3,𝔽₂)⋊C₃	C₂³
Fitting	C₃²×C₂²⋊C₄	GL(2,𝔽₃)
socle	C₂²×C₃²	C₂²×GL(2,𝔽₃)
hypercenter	C₂²⋊C₄	AGL(2,𝔽₃)
hypocenter	PSU(3,𝔽₂)⋊C₃	#120
LCS	G ▷ C₂²×PSU(3,𝔽₂)⋊C₃ ▷ PSU(3,𝔽₂)⋊C₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

3022 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 3022 conjugacy classes of subgroups — too many to draw. Only its 86 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:

2^5,3,2^3,3^2|a^2=d,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=ce,f^b=f^2,g^b=gi,g^f=h,h^b=gh,h^f=ghi,h^g=hi,j^b=k,j^f=k,j^g=k,j^h=jk^2,j^i=j^2,k^b=j,k^f=j^2k^2,k^g=j^2,k^h=j^2k^2,k^i=k^2

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#120×AGL(2,𝔽₃), #72252985
maximal normal subgroup of	#120×AGL(2,𝔽₃), #72252985
maximal quotient of	#72252985

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 6, F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.11);;
gap> SetConjugate(coll, 10, 6, F.11);;
gap> SetConjugate(coll, 10, 7, F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11^2);;
gap> SetConjugate(coll, 10, 9, F.10^2);;
gap> SetConjugate(coll, 11, 2, F.10);;
gap> SetConjugate(coll, 11, 6, F.10^2*F.11^2);;
gap> SetConjugate(coll, 11, 7, F.10^2);;
gap> SetConjugate(coll, 11, 8, F.10^2*F.11^2);;
gap> SetConjugate(coll, 11, 9, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);