#72233455

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂⁴×C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#6342371 ⋊ C₃²
derived	C₂⁸ ⋊ C₂×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂⁸	C₂×C₃²
Frattini, socle	C₂⁴	C₂×A₄²
Fitting	#6342371	C₃²
LCS	G ▷ C₂⁸
UCS	C₁

24577 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 24577 conjugacy classes of subgroups — too many to draw. Only its 29 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 — 2 generators on 16 points:

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

Polycyclic presentation:

2,3^2,2^8|d^a=dhi,d^b=e,d^c=e,e^a=eh,e^b=de,e^c=de,f^a=fj,f^b=g,f^c=fhi,g^a=gk,g^b=fg,g^c=gh,h^b=i,h^c=i,i^b=hi,i^c=hi,j^b=k,k^b=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258349, #72258415, #72258594
maximal normal subgroup of	#72258349, #72258415, #72258594

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