#72247489

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Hall	C₅×#737 ⋊ C₂×C₄
Sylow	#737 ⋊ C₂×AGL(1,𝔽₅)
Sylow	C₅ ⋊ #4207⋊C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, hypocenter	C₅×#737	C₂×C₄
Frattini	C₃	#11762163
socle	C₃×C₅	#6294
prime radical	D₅×#4207	C₂
cosocle	D₅×#737	C₂²
LCS	G ▷ C₅×#737
UCS	C₁

234 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 135

Polycyclic presentation:

2^3,3^3,5,3|a^2=c,e^3=h,d^a=d^2,d^b=d^2,e^b=e^2h^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)×#4207⋊C₂, #72254469
maximal normal subgroup of	AGL(1,𝔽₅)×#4207⋊C₂, #72254469

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> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.5^2*F.8^2);;
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);