#72247498

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

Aliases

No aliases for this group.

Splittings

Fitting, derived	C₂²×C₅×C₉ ⋊ C₃×C₄
Sylow	C₅ ⋊ Dic₉⋊A₄

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, hypocenter	C₂²×C₅×C₉	C₃×C₄
Frattini	C₃	A₄×ΓL(1,𝔽₁₆)
socle	C₂²×C₃×C₅	C₃×Dic₃
prime radical	C₃×D₅×A₄	D₃
cosocle	C₂²×C₉×D₅	C₂×C₃
LCS	G ▷ C₂²×C₅×C₉
UCS	C₁

122 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 27 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 — 2 generators on 18 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄×3₋¹⁺²⋊AGL(1,𝔽₅), AGL(1,𝔽₅)×D₉⋊A₄
maximal normal subgroup of	A₄×3₋¹⁺²⋊AGL(1,𝔽₅), AGL(1,𝔽₅)×D₉⋊A₄

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