#72228099

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	C₄²×#127 ⋊ C₃
derived	C₄⁴ ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₄⁴	C₂×C₃
Frattini, socle	C₂⁴	C₂×C₂²⋊A₄
Fitting	C₄²×#127	C₃
LCS	G ▷ C₄⁴
UCS	C₁

2290 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 2290 conjugacy classes of subgroups — too many to draw. Only its 35 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) = 24
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,5)(3,8,10)(4,6,11)(7,12,9)(13,14,18)(15,21,22,16,20,24)(17,19,23);(1,9,4,3)(2,7,6,8)(5,11)(10,12)(13,16,17,15)(14,19)(18,24,23,22)(20,21);(14,20,19,21)(18,24,23,22)

Polycyclic presentation:

2,3,2^8|c^2=g,d^2=h,e^2=i,f^2=j,c^a=cg,c^b=cd,d^a=dh,d^b=ch,e^a=egh,e^b=fi,f^a=fh,f^b=ef,g^b=gh,h^b=g,i^b=j,j^b=ij

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄²⋊C₃×#127⋊C₃, #72228270, #72258805
maximal normal subgroup of	C₄²⋊C₃×#127⋊C₃, #72228270, #72258805

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