#72220119

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting

C₃²×C₈ ⋊ GL(2,𝔽₃)

Subgroups

	subgroup	G/H
center	C₂	C₄⋊AGL(2,𝔽₃)
commutator, cosocle	C₄×PSU(3,𝔽₂)⋊C₃	C₂²
Frattini	C₄	C₂×AGL(2,𝔽₃)
Fitting	C₃²×C₈	GL(2,𝔽₃)
socle	C₂×C₃²	C₄⋊GL(2,𝔽₃)
prime radical	C₄⋊AGL(2,𝔽₃)	C₂
hypercenter	C₈	AGL(2,𝔽₃)
hypocenter	PSU(3,𝔽₂)⋊C₃	SD₁₆
LCS	G ▷ C₄×PSU(3,𝔽₂)⋊C₃ ▷ C₂×PSU(3,𝔽₂)⋊C₃ ▷ PSU(3,𝔽₂)⋊C₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

324 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 26 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:

2^4,3,2^3,3^2|a^2=c,c^2=d,f^2=h,g^2=h,b^a=bcd,c^b=cd,e^b=e^2,f^b=gh,f^e=g,g^b=fh,g^e=fgh,g^f=gh,i^b=j,i^e=j^2,i^f=ij,i^g=ij^2,i^h=i^2,j^b=i,j^e=ij^2,j^f=ij^2,j^g=i^2j^2,j^h=j^2

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₁₆×AGL(2,𝔽₃), #72220192, #72220196, …
maximal normal subgroup of	SD₁₆×AGL(2,𝔽₃), #72220192, #72220196, …
maximal quotient of	#72245046

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