#72245011

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₂×C₃²

solvable

Aliases

No aliases for this group.

Splittings

Sylow

#585 ⋊ C₃×3₋¹⁺²

Subgroups

	subgroup	G/H
center	C₃×C₄	C₃×A₄²
commutator, cosocle	C₃×H(5,𝔽₂)	C₂×C₃³
Frattini, socle	C₂×C₃	C₂×C₃×A₄²
Fitting	C₃²×#585	C₃²
prime radical	C₃²×#5950	C₃
hypercenter	C₃²×C₄	A₄²
hypocenter	H(5,𝔽₂)	C₂×C₃×3₋¹⁺²
LCS	G ▷ C₃×H(5,𝔽₂) ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₃×C₄ ◁ C₃²×C₄

810 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 101 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) = 144

Polycyclic presentation:

2,3^4,2^5|a^2=j,d^3=e,f^2=j,g^2=j,d^b=de,f^c=g,f^d=g,g^c=fgj,g^d=fgj,g^f=gj,h^c=i,h^d=fgh,h^g=hj,i^c=hi,i^d=fij,i^f=ij

Conjugacy classes & character table

194 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245618, #72245689, #72245730
maximal normal subgroup of	#72245618, #72245689, #72245730

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