Finite Group Databasebeta

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#72341747

order3072
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Splittings

FittingC4×#3027 ⋊ D3

Subgroups

subgroupG/H
center, socleC22C2×#73081
commutator#5950C22×C4
FrattiniC23×C4C22×S4
FittingC4×#3027D3
prime radicalC2×#11711638C2
cosocleC2×#5950C23
hypercenterC42#1020⋊C2
hypocenter#997C4×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C42

3064 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 3064 conjugacy classes of subgroups — too many to draw. Only its 69 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)(5,35,20,9,24,18,8,34,11,7,25,22)(6,36,26,12,33,30,14,31,16,13,32,28)(10,17,19,21)(15,29,23,27);(6,12)(7,9)(10,24)(13,14)(15,31)(16,28)(17,34)(18,22)(19,25)(21,35)(23,36)(26,30)(27,33)(29,32);(1,2,4,3)(5,6,8,14)(7,13,9,12)(10,23,19,15)(11,26,20,16)(17,27,21,29)(18,28,22,30)(24,32,25,33)(31,34,36,35)

Polycyclic presentation:

2^5,3,2^5|a^2=d,c^2=d,e^2=k,g^2=k,h^2=k,b^a=be,c^b=ck,e^a=ek,e^b=ek,f^b=f^2,g^a=gijk,g^b=ghjk,g^f=h,h^a=hik,h^b=hi,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

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341808, #72342093
maximal normal subgroup of#72341808, #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, 1, F.4);;
gap> SetPower(coll, 3, F.4);;
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, 2, F.3*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 2, 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, 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, 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);