#72247667

order	2160
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₅×C₉ ⋊ Dic₃
derived	C₅×C₉×A₄ ⋊ C₄
Sylow	C₅ ⋊ C₉⋊SL(2,ℤ/4)

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₅×C₉×A₄	C₄
Frattini	C₃	C₃⋊(A₄⋊AGL(1,𝔽₅))
Fitting	C₂²×C₅×C₉	Dic₃
socle	C₂²×C₃×C₅	C₃⋊Dic₃
prime radical	C₃×D₅×A₄	D₃
cosocle	C₉×D₅×A₄	C₂
LCS	G ▷ C₅×C₉×A₄
UCS	C₁

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

normal subgroups only

This group has 162 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

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 180

A minimal faithful permutation representation — 3 generators on 18 points:

(1,4,3,2)(6,7,8,9)(11,13)(12,16)(14,17)(15,18);(5,6,9,7,8)(10,15,13,12,17,14,16,11,18);(2,3,4)

Polycyclic presentation:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72247686, S₄×C₉⋊AGL(1,𝔽₅), D₉×A₄⋊AGL(1,𝔽₅), …
maximal normal subgroup of	#72247686, S₄×C₉⋊AGL(1,𝔽₅), D₉×A₄⋊AGL(1,𝔽₅), …

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