Finite Group Databasebeta

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#72341862

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

Aliases

No aliases for this group.

Splittings

Fitting#6372734 ⋊ D3

Subgroups

subgroupG/H
center, socleC22#11713201
commutator#5950C22×C4
FrattiniC23×C4C22×S4
Fitting#6372734D3
prime radicalC2×#11711638C2
cosocleC2×#5950C23
hypercenterC4⋊C4#1020⋊C2
hypocenter#997C4×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

2581 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 2581 conjugacy classes of subgroups — too many to draw. Only its 67 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:

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

Polycyclic presentation:

2^5,3,2^5|a^2=d,e^2=k,g^2=k,h^2=k,c^a=ck,c^b=ce,e^b=ek,e^c=ek,f^b=f^2,g^a=gijk,g^b=ghjk,g^c=gijk,g^f=h,h^a=hik,h^b=hi,h^c=hik,h^f=ghk,h^g=hk,i^f=j,i^g=ik,j^b=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#72341940
maximal normal subgroup of#72341940

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