#72233211

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁴×#120 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	C₂×#87588⋊C₂
commutator, cosocle	C₂×C₂²⋊(C₂²⋊A₄)	C₂³
Frattini	C₂³	C₂²×#1020
Fitting	C₂⁴×#120	D₃
socle	C₂⁵	C₂²×S₄
hypercenter	C₂²	#87588⋊C₂
hypocenter	C₂²⋊(C₂²⋊A₄)	C₂×D₄
LCS	G ▷ C₂×C₂²⋊(C₂²⋊A₄) ▷ C₂²⋊(C₂²⋊A₄)
UCS	C₁ ◁ C₂ ◁ C₂²

At least 8832 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has at least 8832 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 24

A minimal faithful permutation representation — 3 generators on 16 points:

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

Polycyclic presentation:

2^4,3,2^6|c^b=cd,e^c=e^2,f^c=g,f^e=g,g^c=f,g^e=fg,h^a=hjk,h^c=i,h^e=i,i^a=ij,i^c=h,i^e=hi,j^c=jk,j^e=k,k^e=jk

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235010, #72240913, #72255861
maximal normal subgroup of	#72235010, #72240913
maximal quotient of	#72256454
automorphism group of	#1655

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