#72220197

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×D₄ ⋊ SD₁₆
Sylow	C₃² ⋊ #18758

Subgroups

	subgroup	G/H
center	C₂	C₂×#10596418
commutator, cosocle	C₄×#157	C₂⁴
Frattini	C₄	C₂²×AΓL(1,𝔽₉)
Fitting	C₂×C₃²×D₄	SD₁₆
socle	C₂×C₃²	C₂×#448
prime radical	#11865897	C₂
hypercenter	C₂×D₄	AΓL(1,𝔽₉)
hypocenter	C₃²	#18758
LCS	G ▷ C₄×#157 ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

1341 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1341 conjugacy classes of subgroups — too many to draw. Only its 142 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:

2^8,3^2|a^2=e,d^2=h,e^2=g,f^2=h,b^a=beg,c^a=cg,d^a=df,e^b=eg,f^a=fh,f^d=fh,i^a=i^2,i^d=ij,i^f=i^2j,i^h=i^2,j^a=ij,j^d=ij^2,j^f=ij,j^h=j^2

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220199, AGL(1,ℤ/8)×AΓL(1,𝔽₉)
maximal normal subgroup of	AGL(1,ℤ/8)×AΓL(1,𝔽₉)

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