#72258530

order	15360
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#62817 ⋊ Dic₃
derived	C₅×#11710730 ⋊ C₄
Sylow	C₅ ⋊ #72245233

Subgroups

	subgroup	G/H
center	C₂	#72246748
commutator, hypocenter	C₅×#11710730	C₄
Frattini, hypercenter	C₂²	#72246556
Fitting	C₅×#62817	Dic₃
socle	C₂×C₅	#72219111
prime radical, cosocle	D₅×#11710730	C₂
LCS	G ▷ C₅×#11710730
UCS	C₁ ◁ C₂ ◁ C₂²

At least 7516 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has at least 7516 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

(2,3,4,5)(6,12,14,17)(7,9,18,15)(8,13,16,21)(10,19,20,11);(1,2,5,3,4)(8,9,14)(11,12,18)(15,16,20)(17,19,21);(8,15)(11,17)(12,19)(14,20)

Polycyclic presentation:

2^2,3,5,2^8|a^2=b,c^a=c^2,d^a=d^3,d^b=d^4,e^a=efi,e^c=f,f^a=fj,f^c=ef,g^a=h,g^c=h,g^e=gk,g^f=gk,h^a=g,h^c=gh,h^f=hk,i^a=ij,i^c=ij,i^h=il,j^c=i,j^g=jl,j^h=jl,k^a=kl

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 3, 5, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^3);;
gap> SetConjugate(coll, 4, 2, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.9);;
gap> SetConjugate(coll, 5, 3, F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 3, F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.12);;
gap> SetConjugate(coll, 10, 3, F.9);;
gap> SetConjugate(coll, 10, 7, F.10*F.12);;
gap> SetConjugate(coll, 10, 8, F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);