#72267932

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#403355 ⋊ C₃

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120⋊C₃
commutator, cosocle	C₂×#543	C₂²×C₃
Frattini	C₂⁵	C₂²×A₄
Fitting	#403355	C₃
hypercenter	C₂⁴	#120⋊C₃
hypocenter	#543	C₃×D₄
LCS	G ▷ C₂×#543 ▷ #543
UCS	C₁ ◁ C₂³ ◁ C₂⁴

1852 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 1852 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 64

A minimal faithful permutation representation — 2 generators on 20 points:

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

Polycyclic presentation:

2^2,3,2^7|e^2=i,f^2=i,b^a=bd,e^a=egh,e^c=f,f^a=fg,f^c=efi,f^e=fi,g^c=hij,g^e=gij,h^c=ghij,h^e=hij,h^f=hij

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72267936, #72268028, #72268096, …

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