#72254802

order	4320
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

derived, Hall	C₃³×C₅ ⋊ C₂×C₄²
Sylow	C₃³ ⋊ C₂×C₄×AGL(1,𝔽₅)
Sylow	C₅ ⋊ Dic₃×#624

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72222379
commutator, hypocenter	C₃³×C₅	C₂×C₄²
Fitting, socle	C₂×C₃³×C₅	C₂²×C₄
prime radical	C₂×C₃×D₅×C₃⋊D₃	C₂²
cosocle	C₂×C₃³×D₅	C₂³
LCS	G ▷ C₃³×C₅
UCS	C₁ ◁ C₂

1124 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 120

Polycyclic presentation:

2^5,3^3,5|a^2=d,b^2=de,f^a=f^2,f^c=f^2,g^a=f^2g,g^c=g^2,h^b=h^2,i^a=i^3,i^b=i^3,i^d=i^4

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₅)×#624
maximal normal subgroup of	D₃×AGL(1,𝔽₅)×#624

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