#72220559

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₃⁴ ⋊ C₂×C₂²⋊C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	(#157⋊D₃)⋊D₃
commutator	C₃²×C₃⋊D₃	C₂²×C₄
Fitting, socle	C₂×C₃⁴	C₂×D₄
prime radical	C₂×#10620215	C₂
cosocle	C₂×C₃²×C₃⋊D₃	C₂³
hypocenter	C₃⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₃²×C₃⋊D₃ ▷ C₃⁴
UCS	C₁ ◁ C₂

1290 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 1290 conjugacy classes of subgroups — too many to draw. Only its 77 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|a^2=e,b^a=bd,f^a=fg,f^b=f^2g^2,f^c=f^2,f^d=f^2,f^e=f^2,g^a=fg^2,g^c=g^2,g^d=g^2,g^e=g^2,h^a=fh,h^b=h^2,h^c=h^2,h^d=f^2gh,h^e=f^2gh,i^a=gi,i^b=h^2i,i^c=i^2,i^d=fi,i^e=fi

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220560, #72220706, #72220719
maximal normal subgroup of	#72220560, #72220706, #72220719

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