#72247374

order	3600
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Fitting, derived, Hall	C₅²×C₉ ⋊ C₄²
Sylow	C₅² ⋊ C₄×Dic₉
Sylow	C₉ ⋊ AGL(1,𝔽₅)²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, hypocenter	C₅²×C₉	C₄²
Frattini	C₃	ΓL(1,𝔽₁₆)⋊AGL(1,𝔽₅)
socle	C₃×C₅²	C₄×Dic₃
prime radical	C₃×D₅²	C₂×D₃
cosocle	C₉×D₅²	C₂²
LCS	G ▷ C₅²×C₉
UCS	C₁

189 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 180

Polycyclic presentation:

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

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×AGL(1,𝔽₅)², #72247437
maximal normal subgroup of	D₉×AGL(1,𝔽₅)², #72247437

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