#72229319

order	3600
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₅² ⋊ C₄⋊C₄
Sylow	C₅² ⋊ C₄⋊(C₃⋊Dic₃)
Sylow	C₃² ⋊ #87832

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅⋊(C₃⋊ΓL(1,𝔽₁₆))
commutator, Fitting, socle	C₂×C₃²×C₅²	C₂×C₄
prime radical, cosocle	C₂×C₃²×C₅⋊D₅	C₂²
hypocenter	C₃²×C₅²	C₄⋊C₄
LCS	G ▷ C₂×C₃²×C₅² ▷ C₃²×C₅²
UCS	C₁ ◁ C₂

408 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 20
transitive degree	μ_t(G) = 180

A minimal faithful permutation representation — 3 generators on 20 points:

(2,4)(6,7,8,9)(10,11,12,13)(16,17)(19,20);(1,2,3,4)(5,6,9,7,8)(11,12)(13,14)(18,19,20);(15,16,17)

Polycyclic presentation:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231471, C₃⋊D₃×#87832, #72231476, …
maximal normal subgroup of	#72231471, C₃⋊D₃×#87832, #72231476, …
maximal quotient of	C₂×#72229319

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