#72252932

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ #8175

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×AΓL(1,𝔽₉)
commutator	C₂×#157	C₂×C₄²
Fitting	C₃²×C₄²	SD₁₆
socle	C₂²×C₃²	C₂×#103
prime radical	C₂²×C₃⋊D₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×#157	C₂³
hypercenter	C₄²	AΓL(1,𝔽₉)
hypocenter	C₃²	#8175
LCS	G ▷ C₂×#157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

651 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 651 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 288

Polycyclic presentation:

2^8,3^2|a^2=d,b^2=e,c^2=h,e^2=g,f^2=h,b^a=bg,c^b=cf,f^b=fh,f^c=fh,i^b=i^2,i^c=ij,i^f=i^2j,i^h=i^2,j^b=ij,j^c=ij^2,j^f=ij,j^h=j^2

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252936, #97×AΓL(1,𝔽₉)
maximal normal subgroup of	#97×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.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 9, 2, 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, 2, 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);