Finite Group Databasebeta

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#72252975

order15552
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3

Aliases

No aliases for this group.

Splittings

derived#72244951 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2#72220160
commutator, cosocle, hypocenter#72244951C2×C3
Frattini, socleC2×C32C3×A4⋊S4
FittingC33×H(5,𝔽2)C3⋊D3
LCSG ▷ #72244951
UCSC1 ◁ C2

1338 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 1338 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) = 42
transitive degreeμt(G) = 648

A minimal faithful permutation representation — 3 generators on 42 points:

(1,11,6,3)(2,22,15,7)(4,14)(5,13)(8,24,9,17)(10,18,23,20)(12,21,16,19)(26,32,33,28,29,30)(27,31)(35,41,42,37,38,39)(36,40);(3,13,14)(4,11,5)(8,23,12)(9,16,15)(17,19,22)(18,21,24)(34,38,37,36,42,39,40,35,41);(1,2,7)(3,12,19)(4,15,21)(5,16,18)(6,10,20)(8,22,13)(9,24,11)(14,23,17)(25,29,28,27,33,30,31,26,32)

Polycyclic presentation:

2,3^5,2^5|c^3=e,d^3=f,g^2=k,h^2=k,c^a=c^2e^2,c^b=ce,d^a=d^2f^2,d^b=df,e^a=e^2,f^a=f^2,g^a=gk,g^c=hk,g^d=ghk,h^a=ghk,h^c=gh,h^d=gk,h^g=hk,i^a=gi,i^c=gi,i^d=j,i^g=ik,j^a=hijk,j^c=ghjk,j^d=ij,j^h=jk

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

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