#72215280

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×H(5,𝔽₂) ⋊ D₃²
derived	Ω⁺(4,𝔽₃) ⋊ C₂³

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂	C₂×S₄²
commutator, cosocle, hypocenter	Ω⁺(4,𝔽₃)	C₂³
Fitting	C₂×H(5,𝔽₂)	D₃²
hypercenter	C₂²	S₄²
LCS	G ▷ Ω⁺(4,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₂²

1475 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 1475 conjugacy classes of subgroups — too many to draw. Only its 45 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:

2^3,3^2,2^5|a^2=j,f^2=j,g^2=j,c^a=cj,d^a=d^2,d^b=d^2,e^a=d^2e,e^b=e^2,f^a=fj,f^b=fj,f^d=gj,f^e=fgj,g^a=fgj,g^b=fgj,g^d=fg,g^e=fj,g^f=gj,h^a=fhj,h^b=fh,h^d=fh,h^e=i,h^f=hj,i^a=ij,i^b=ghij,i^d=fgij,i^e=hi,i^g=ij

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72215280, #72228062, C₃×#72215280, …
maximal normal subgroup of	C₂×#72215280, #72228062, C₃×#72215280
maximal quotient of	#72226130, #72226221, #72226236, …
holomorph of	#226, GL(2,𝔽₃)

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, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.4^2*F.5);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 1, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.6*F.8);;
gap> SetConjugate(coll, 8, 4, F.6*F.8);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);