#72244273

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₉

Aliases

No aliases for this group.

Splittings

Fitting	C₉²×D₈ ⋊ C₂
Sylow	C₉² ⋊ AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	D₄×C₉⋊D₉
commutator, cosocle	C₄×C₉²	C₂³
Frattini	C₃²×C₄	C₂²×C₃⋊D₃
Fitting	C₉²×D₈	C₂
socle	C₂×C₃²	D₄×C₃⋊D₃
hypercenter	D₈	C₉⋊D₉
hypocenter	C₉²	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₉² ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

782 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 648

Polycyclic presentation:

2^5,3^4|c^2=d,d^2=e,f^3=h,g^3=i,c^a=cd,c^b=ce,d^a=de,f^a=f^2h^2,f^b=f^2h^2,g^a=g^2i^2,g^b=g^2i^2,h^a=h^2,h^b=h^2,i^a=i^2,i^b=i^2

Conjugacy classes & character table

291 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×C₉⋊D₉, C₂×#72244273
maximal normal subgroup of	AGL(1,ℤ/8)×C₉⋊D₉, C₂×#72244273
maximal quotient of	#72244280, #72254568, C₂×#72244273

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