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#72342029

order3072
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#11713200
commutator#5950C22×C4
FrattiniC23×C4C22×S4
FittingC2×#32894D3
prime radicalC2×#32894⋊C3C2
cosocleC2×#5950C23
hypercenterC2×D4#1020⋊C2
hypocenter#997C4×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

5390 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 5390 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 36
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(1,3,4,2)(5,35,24,18)(6,31,30,25)(7,23,19,10)(8,26,17,27)(9,20,11,22)(12,36,32,29)(13,21,14,15)(16,34,28,33);(1,2,4,3)(5,6,8,12,11,13,7,16)(9,21,19,36,24,31,17,34)(10,25,20,33,27,15,18,29)(14,22,28,26,30,35,32,23);(6,12)(7,8)(13,16)(14,28)(15,29)(17,19)(18,20)(21,36)(22,35)(25,33)(30,32)(31,34)

Polycyclic presentation:

2^5,3,2^5|b^2=d,c^2=dek,d^2=k,e^2=k,g^2=k,h^2=k,b^a=be,c^a=ck,c^b=ce,d^a=dk,d^c=dk,e^a=ek,e^c=ek,f^b=f^2,f^c=f^2,g^b=gh,g^c=ghjk,g^f=h,h^b=hk,h^c=hi,h^f=ghk,h^g=hk,i^f=j,i^g=ik,j^b=ij,j^c=ij,j^f=ij,j^g=jk,j^h=jk

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342093
maximal normal subgroup of#72342093

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