#72220535

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×D₄ ⋊ SD₁₆
Sylow	C₃² ⋊ #28482

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×AΓL(1,𝔽₉)
commutator, cosocle	C₂²×#157	C₂⁴
Fitting	C₂×C₃²×D₄	SD₁₆
socle	C₂²×C₃²	C₂²×SD₁₆
prime radical	#11824465	C₂
hypercenter	C₂×D₄	AΓL(1,𝔽₉)
hypocenter	C₃²	#28482
LCS	G ▷ C₂²×#157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×D₄

1511 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1511 conjugacy classes of subgroups — too many to draw. Only its 170 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|b^2=g,d^2=fh,g^2=h,c^a=ce,d^a=df,d^b=dgh,d^c=df,g^d=gh,i^b=i^2j^2,i^d=ij,i^g=i^2j,i^h=i^2,j^b=i^2,j^d=ij^2,j^g=ij,j^h=j^2

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#127×AΓL(1,𝔽₉)
maximal normal subgroup of	#127×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, 2, F.7);;
gap> SetPower(coll, 4, F.6*F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.9^2*F.10^2);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9^2*F.10);;
gap> SetConjugate(coll, 9, 8, F.9^2);;
gap> SetConjugate(coll, 10, 2, F.9^2);;
gap> SetConjugate(coll, 10, 4, F.9*F.10^2);;
gap> SetConjugate(coll, 10, 7, F.9*F.10);;
gap> SetConjugate(coll, 10, 8, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);