#72227648

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

H(9,𝔽₂) ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂	#72209718
commutator, cosocle	C₂×#5950	C₂³
Frattini	C₂×C₄	C₂²×#1020
Fitting	H(9,𝔽₂)	D₃
hypercenter	H(5,𝔽₂)	#1020
hypocenter	#997	#120
LCS	G ▷ C₂×#5950 ▷ #997
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

At least 9395 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has at least 9395 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,4,5,16,6,17,18,15)(2,19,11,32,8,23,22,25)(3,28,14,31,12,13,29,30)(7,10,20,26,9,21,24,27);(1,2)(3,7)(4,26,5,10,16,11)(6,8)(9,12)(13,32,14,19,30,20)(15,22,17,27,18,21)(23,31,24,28,25,29);(3,12)(7,9)(13,28)(14,29)(19,23)(20,24)(25,32)(30,31)

Polycyclic presentation:

2^5,3,2^5|e^2=k,i^2=k,j^2=k,c^a=cd,c^b=ce,d^b=dk,e^a=ek,e^b=ek,e^c=ek,f^c=f^2,g^c=gh,g^f=h,h^f=gh,i^c=ij,i^f=j,i^h=ik,j^c=jk,j^f=ijk,j^g=jk,j^i=jk

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227651, #72252511, #72267056
maximal normal subgroup of	#72252511, #72267056
maximal quotient of	#72268143, #72268261, #72269004, …

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