Finite Group Databasebeta

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#72235302

order6144
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Splittings

derived#11709219 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenter#11709219C2×C4
FrattiniC25C2×#1020
Fitting#72234510D3
socleC24C2×#5936
prime radical#72234946C2
cosocle#72234945C22
LCSG ▷ #11709219
UCSC1

4685 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 4685 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^3,3,2^8|b^2=c,g^2=i,h^2=j,b^a=bk,d^a=dk,d^b=d^2,e^a=eikl,e^b=fik,e^c=eik,e^d=f,f^a=fjk,f^b=eijkl,f^c=fjl,f^d=ef,g^a=gij,g^b=fhkl,g^c=gj,g^d=hi,g^e=gl,g^f=gk,h^a=hi,h^b=egik,h^c=hij,h^d=gh,h^e=hk,h^f=hkl,i^b=jkl,i^d=j,j^b=il,j^d=ij,k^d=l,l^b=kl,l^d=kl

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235323

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.11*F.12);;
gap> SetConjugate(coll, 5, 2, F.6*F.9*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.11);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.5*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.10*F.12);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.6*F.8*F.11*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.12);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.5*F.7*F.9*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.11*F.12);;
gap> SetConjugate(coll, 9, 2, F.10*F.11*F.12);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 2, F.9*F.12);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 11, 4, F.12);;
gap> SetConjugate(coll, 12, 2, F.11*F.12);;
gap> SetConjugate(coll, 12, 4, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);