#72229634

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₅ ⋊ #440
Sylow	C₃² ⋊ #65735
Sylow	C₅ ⋊ #10591446

Subgroups

	subgroup	G/H
center	C₂	#67⋊ΓL(1,𝔽₁₆)
commutator	C₃²×C₄×C₅	C₂²×C₄
Frattini	C₄	C₂×D₃×ΓL(1,𝔽₁₆)
Fitting	C₃²×C₅×D₄	C₂×C₄
socle	C₂×C₃²×C₅	C₄×D₄
prime radical	C₃²×D₄×D₅	C₂²
cosocle	C₃²×C₄×D₅	C₂³
hypercenter	D₄	D₃×ΓL(1,𝔽₁₆)
hypocenter	C₃²×C₅	#440
LCS	G ▷ C₃²×C₄×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 3 generators on 19 points:

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

Polycyclic presentation:

2^6,3^2,5|a^2=e,c^2=d,e^2=f,b^a=bef,c^a=cf,e^b=ef,g^a=g^2,g^c=g^2,h^a=g^2h,h^c=h^2,i^c=i^3,i^d=i^4

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230434, #72230457, AGL(1,𝔽₅)×#63937
maximal normal subgroup of	#72230434, #72230457, AGL(1,𝔽₅)×#63937

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 3, 5]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 3, F.9^3);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);