#72220323

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₃²

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃⁴ ⋊ C₂×C₄
Sylow	C₃⁴ ⋊ C₂×C₂²⋊C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#10620198
commutator, socle	C₂×C₃⁴	C₂²×C₄
Fitting	C₂²×C₃⁴	C₂×C₄
prime radical	C₂²×C₃⋊D₃²	C₂
cosocle	C₂×C₃⋊(C₃⋊(C₃⋊D₃))	C₂³
hypercenter	C₂²	#10620198
hypocenter	C₃⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

2112 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 2112 conjugacy classes of subgroups — too many to draw. Only its 57 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=d,b^a=be,f^a=fh,f^c=f^2,f^d=f^2,g^a=gi,g^c=f^2g,g^d=g^2,h^a=fh^2,h^c=h^2,h^d=h^2,i^a=gi^2,i^c=h^2i,i^d=i^2

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220568, #72220759, #72258373
maximal normal subgroup of	#72220568, #72220759, #72258373
maximal quotient of	#72220324, #72220331

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