#72213220

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting, socle, Sylow

C₃⁴ ⋊ #120

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₃⋊D₃²	C₂³
Fitting, socle, hypocenter	C₃⁴	#120
LCS	G ▷ C₃⋊D₃² ▷ C₃⁴
UCS	C₁

1029 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 1029 conjugacy classes of subgroups — too many to draw. Only its 33 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:

2^5,3^4|b^a=be,c^b=cd,f^a=fh,f^b=f^2,f^c=fh,f^d=f^2,f^e=f^2,g^a=gh,g^b=g^2,g^c=gi,g^d=g^2,g^e=fg,h^a=h^2,h^b=f^2h,h^c=h^2,h^d=h^2,h^e=h^2,i^a=hi,i^b=g^2i,i^c=i^2,i^d=i^2,i^e=hi

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	GO⁺(4,𝔽₂)²
maximal normal subgroup of	GO⁺(4,𝔽₂)²
maximal quotient of	#72260117
holomorph of	D₃²

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