Finite Group Databasebeta

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#72247682

order9216
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

derived#11710729 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC2#72215845
commutator, hypocenter#11710729C3×C4
Frattini, socleC23#1020⋊A4
FittingC2×#62817C3×D3
prime radicalC2×#72215828C2
cosocleC2×#11710729C2×C3
LCSG ▷ #11710729
UCSC1 ◁ C2

4643 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 4643 conjugacy classes of subgroups — too many to draw. Only its 21 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) = 20
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 20 points:

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

Polycyclic presentation:

2,3,2^3,3,2^6|a^2=c,d^b=de,e^b=d,f^a=f^2,g^b=ghi,g^d=gl,g^f=j,h^a=i,h^b=gj,h^d=hk,h^e=hk,h^f=hi,i^a=h,i^b=j,i^d=ikl,i^e=ikl,i^f=h,j^a=gj,j^b=ij,j^d=jkl,j^f=gj,k^a=kl,k^f=l,l^f=kl

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#72215845, #72248221, #72253698
maximal normal subgroup ofC4×#72215845, #72248221, #72253698

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", "l");;
gap> coll := SingleCollector(F, [2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.4);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 7, 6, F.10);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.8);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 5, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 6, F.8);;
gap> SetConjugate(coll, 10, 1, F.7*F.10);;
gap> SetConjugate(coll, 10, 2, F.9*F.10);;
gap> SetConjugate(coll, 10, 4, F.10*F.11*F.12);;
gap> SetConjugate(coll, 10, 6, F.7*F.10);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 6, F.12);;
gap> SetConjugate(coll, 12, 6, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);