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#72246265

order3072
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Splittings

Fitting#6591514 ⋊ D3

Subgroups

subgroupG/H
center, socleC22C22×#1020⋊C2
commutator, cosocleC2×#997C24
FrattiniC24C23×S4
Fitting#6591514D3
hypercenterC2×D4#1020⋊C2
hypocenter#997H(5,𝔽2)
LCSG ▷ C2×#997 ▷ #997
UCSC1 ◁ C22 ◁ C2×D4

6862 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 6862 conjugacy classes of subgroups — too many to draw. Only its 113 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) = 24
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^5,3,2^5|d^2=ek,g^2=k,h^2=k,b^a=be,c^a=cek,c^b=cek,d^a=dek,d^b=dek,d^c=dek,f^d=f^2,g^c=gik,g^d=gh,g^f=h,h^c=hijk,h^d=hk,h^f=ghk,h^g=hk,i^d=j,i^f=ij,i^h=ik,j^d=i,j^f=i,j^g=jk,j^h=jk

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72247039, C2×#72246265
maximal normal subgroup of#72247039, C2×#72246265
maximal quotient ofC2×#72246265

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