#72254528

order	3240
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, derived, Hall	C₃×C₅×H(3,𝔽₃) ⋊ C₂×C₄
Sylow	C₃×H(3,𝔽₃) ⋊ C₂×AGL(1,𝔽₅)
Sylow	C₅ ⋊ #10619883

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, hypocenter	C₃×C₅×H(3,𝔽₃)	C₂×C₄
Frattini	C₃	C₃⋊D₃×ΓL(1,𝔽₁₆)
socle	C₃²×C₅	D₃×Dic₃
prime radical	D₅×C₃⋊((C₃⋊D₃)⋊C₃)	C₂
cosocle	C₃×D₅×H(3,𝔽₃)	C₂²
LCS	G ▷ C₃×C₅×H(3,𝔽₃)
UCS	C₁

432 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 90

Polycyclic presentation:

2^3,3^3,5,3|a^2=c,d^a=d^2,d^b=d^2,e^a=e^2,e^b=e^2,f^a=f^2,f^d=fh,g^a=g^3,g^c=g^4,h^b=h^2

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×((C₃⋊D₃)⋊C₃)⋊D₃
maximal normal subgroup of	AGL(1,𝔽₅)×((C₃⋊D₃)⋊C₃)⋊D₃
maximal quotient of	#72254531

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