#72227619

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

H(9,𝔽₂) ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂	C₂⁴×C₂²⋊A₄
commutator, cosocle, hypocenter	H(5,𝔽₂)	C₂⁴×C₃
Fitting	H(9,𝔽₂)	C₃
hypercenter	H(5,𝔽₂)	C₂²⋊A₄
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

141290 conjugacy classes of subgroups; only the 537 normal subgroups are shown.

normal subgroups only

This group has 141290 conjugacy classes of subgroups — too many to draw. Only its 537 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) = 32

Polycyclic presentation:

2^4,3,2^5|f^2=j,g^2=j,b^a=bj,c^a=cj,c^b=cj,d^a=dj,f^e=fg,g^e=f,g^f=gj,h^e=hi,h^f=hj,h^g=hj,i^e=h,i^g=ij

Conjugacy classes & character table

131 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227620, #72227648, #72227740, …
maximal normal subgroup of	#72227648, #72227841, #72227958, …
maximal quotient of	C₂×#72227619, #72268063, #72268200, …

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