Finite Group Databasebeta

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#72342003

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

Aliases

No aliases for this group.

Splittings

Fitting#392141 ⋊ D3

Subgroups

subgroupG/H
center, socleC22#11713200
commutator#5950C22×C4
FrattiniC23×C4C22×S4
Fitting#392141D3
prime radicalC2×#11715708C2
cosocleC2×#5950C23
hypercenterC2×D4#1020⋊C2
hypocenter#997C4×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

5749 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 5749 conjugacy classes of subgroups — too many to draw. Only its 65 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)(5,24,11,16)(6,30,15,23)(7,27,17,10)(8,33,19,21)(9,25,22,18)(12,36,29,28)(13,31,26,14)(20,35,34,32);(1,2,4,3)(5,6,8,13)(7,12,9,20)(10,26,25,14,11,30,18,15,27,23,19,31)(16,34,33,28,17,35,21,29,24,32,22,36);(6,12)(7,9)(13,20)(14,28)(15,29)(16,21)(17,22)(23,32)(24,33)(26,34)(30,35)(31,36)

Polycyclic presentation:

2^5,3,2^5|b^2=d,c^2=dk,e^2=k,g^2=k,h^2=k,b^a=be,c^a=ck,c^b=cek,e^a=ek,e^b=ek,e^c=ek,f^c=f^2,g^b=gijk,g^c=ghjk,g^f=h,h^b=hik,h^c=hi,h^f=ghk,h^g=hk,i^f=j,i^g=ik,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.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*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, 2, F.7*F.9*F.10*F.11);;
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.9*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, 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);