#72278455

order	11520
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₅×H(5,𝔽₂) ⋊ C₄×D₃
derived	C₃×C₅×#997 ⋊ C₂×C₄
Sylow	C₅ ⋊ #72277406

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72231163
commutator, hypocenter	C₃×C₅×#997	C₂×C₄
Fitting	C₃×C₅×H(5,𝔽₂)	C₄×D₃
socle	C₂×C₃×C₅	C₄×#1020
prime radical	D₅×C₃⋊#5934	C₂
cosocle	C₃×D₅×#997	C₂²
LCS	G ▷ C₃×C₅×#997
UCS	C₁ ◁ C₂

3386 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 3386 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 16
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

2^3,3^2,5,2^5|a^2=c,b^2=c,d^b=d^2,e^a=e^2,f^a=f^3,f^b=f^3,f^c=f^4,g^a=h,g^e=h,h^a=g,h^e=gh,i^a=j,i^e=j,i^h=ik,j^a=i,j^e=ij,j^g=jk

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₅)×#5934
maximal normal subgroup of	D₃×AGL(1,𝔽₅)×#5934

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", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 5, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^3);;
gap> SetConjugate(coll, 6, 2, F.6^3);;
gap> SetConjugate(coll, 6, 3, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 5, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> SetConjugate(coll, 10, 5, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);