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#72277611

order13122
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C32

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72265144 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC3#72277594
commutator, cosocle, hypocenter#10624730C2×C32
FrattiniC34C32×C3⋊D3
Fitting#72265144C2
socleC32#13091469
LCSG ▷ #10624730
UCSC1 ◁ C3

7362 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 7362 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) = 54

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

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

Polycyclic presentation:

2,3^8|d^3=i,c^a=ci^2,c^b=cghi,d^a=d^2i^2,d^b=df,d^c=di^2,e^a=e^2,e^c=ei,e^d=eg,f^a=f^2h,f^c=fi,f^d=fh,g^a=gi^2,g^d=gi,i^a=i^2

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

automorphism group of#6895

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