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#72340845

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#7667013 ⋊ D3

Subgroups

subgroupG/H
centerC2C2×#11715758
commutator, cosocle, hypocenter#2679⋊C3C23
Frattini, socleC23C22×#1020
Fitting#7667013D3
hypercenterC4#11715758
LCSG ▷ #2679⋊C3
UCSC1 ◁ C2 ◁ C4

3492 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 3492 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 48
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 48 points:

(1,35,6,40,7,28,27,16)(2,13,11,29,12,37,15,38)(3,19,14,10)(4,23,18,24)(5,8)(9,33)(17,25)(20,32,31,39)(21,47,41,46)(22,45,48,30)(26,34)(36,44,43,42);(1,9)(2,4,12,18)(3,27,14,6)(5,31,24,20,25,42,23,44)(7,26)(8,11)(10,45,33,41,19,21,34,30)(13,48,38,46)(15,17)(16,32,28,36)(22,37,47,29)(35,39,40,43);(1,2)(3,8)(4,9)(5,19)(6,11)(7,12)(10,25)(13,35)(14,17)(15,27)(16,38)(18,26)(20,30)(21,31)(22,43)(23,33)(24,34)(28,37)(29,40)(32,46)(36,47)(39,48)(41,44)(42,45)

Polycyclic presentation:

2^3,3,2^7|e^2=j,f^2=jk,b^a=bjk,c^a=cjk,c^b=ci,d^a=dk,d^b=d^2,d^c=d^2,e^a=eij,e^b=efij,e^c=efij,e^d=fij,f^a=fijk,f^b=fi,f^c=fi,f^d=efi,f^e=fi,g^a=gijk,g^b=efhjk,g^c=efhjk,g^d=gh,g^e=gj,g^f=gik,h^a=hij,h^b=eg,h^c=eg,h^d=g,h^e=hjk,h^f=hij,j^b=ik,j^c=ik,j^d=jk,k^b=ij,k^c=ij,k^d=ij

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340855
maximal normal subgroup of#72340855

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