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#72276271

order11664
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

derived#72228399 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72228399C22
FrattiniC33C3⋊D3×S4
FittingC22×C32⋊H(3,𝔽3)C2×D3
socleC22×C3#11757730
LCSG ▷ #72228399
UCSC1

1525 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 1525 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 31
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 31 points:

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

Polycyclic presentation:

2^2,3^6,2^2|e^3=h,c^a=c^2,c^b=c^2,d^a=d^2,d^b=d^2,d^c=dg,e^a=e^2h^2,e^d=ef,f^b=f^2,f^c=fh,g^e=gh^2,h^a=h^2,i^e=j,j^a=ij,j^e=ij

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

automorphism group ofC3⋊((C22⋊C9)⋊C2)

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