Finite Group Databasebeta

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#72342866

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

Aliases

No aliases for this group.

Splittings

Fitting#392717 ⋊ D3

Subgroups

subgroupG/H
center, socleC22#11713271
commutator, cosocleC2×#5950C23
FrattiniC23×C4C22×S4
Fitting#392717D3
hypercenterC2×C8#1020⋊C2
hypocenter#997#120
LCSG ▷ C2×#5950 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

3836 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 3836 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 64

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

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

Polycyclic presentation:

2^5,3,2^5|c^2=dek,e^2=k,c^a=cd,c^b=ce,e^b=ek,f^a=f^2,g^a=gh,g^b=gik,g^f=h,h^b=hjk,h^f=gh,i^a=ij,i^f=j,i^h=ik,j^f=ij,j^g=jk

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342916, #72343985, #72344011
maximal normal subgroup of#72342916, #72343985, #72344011

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