Finite Group Databasebeta

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#72245676

order3456
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC4#11724050
commutator, cosocle, hypocenter#63703C22×C3
Frattini, socleC2×C3C2×C3×#1020
FittingC32×#585D3
LCSG ▷ #63703
UCSC1 ◁ C4

1147 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 1147 conjugacy classes of subgroups — too many to draw. Only its 43 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) = 25
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 2 generators on 25 points:

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

Polycyclic presentation:

2^2,3^3,2^5|a^2=j,d^3=e,f^2=j,g^2=j,d^b=d^2e^2,d^c=de,e^b=e^2,f^b=gj,f^d=g,g^b=fj,g^d=fgj,g^f=gj,h^b=ghi,h^d=fgij,h^f=hj,h^g=hj,i^b=fgij,i^d=ghi,i^f=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245689, #72253113
maximal normal subgroup of#72253113

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