#72220474

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×C₂²⋊C₄ ⋊ SD₁₆
Sylow	C₃² ⋊ #9262

Subgroups

	subgroup	G/H
center	C₂	#11787609
commutator	C₂²×#157	C₂²×C₄
Frattini	C₂³	C₂×AΓL(1,𝔽₉)
Fitting	C₃²×C₂²⋊C₄	SD₁₆
socle	C₂×C₃²	#1880
prime radical	C₂×#64242	C₂²
cosocle	C₂³×#157	C₂³
hypercenter	C₂²⋊C₄	AΓL(1,𝔽₉)
hypocenter	C₃²	#9262
LCS	G ▷ C₂²×#157 ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

855 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 855 conjugacy classes of subgroups — too many to draw. Only its 82 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=g,b^2=d,c^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^b=eg,f^a=fh,f^c=fh,i^a=i^2,i^c=ij,i^f=i^2j,i^h=i^2,j^a=ij,j^c=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	#72220476, #72252956, #72252979, …
maximal normal subgroup of	#72252956, #72252979, #99×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.7);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 3, 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, 3, 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);