#72311952

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

C₂²×#3027 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	C₂×#85645
commutator, cosocle, hypocenter	C₂²×H(5,𝔽₂)	C₂²×C₃
Frattini	C₂³	C₂²×C₂²⋊A₄
Fitting	C₂²×#3027	C₃
socle	C₂³	C₂×#120⋊C₃
hypercenter	C₄	#85645
LCS	G ▷ C₂²×H(5,𝔽₂)
UCS	C₁ ◁ C₂ ◁ C₄

7976 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 7976 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 48

Polycyclic presentation:

2^2,3,2^7|a^2=j,f^2=j,g^2=j,b^a=bj,d^c=e,e^c=de,f^b=fhi,f^c=g,g^b=gh,g^c=fgj,g^f=gj,h^c=ij,h^f=hj,i^c=hij,i^f=ij,i^g=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311957, #72311965, #72312025
maximal normal subgroup of	#72311957, #72311965, #72312025

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