Finite Group Databasebeta

← Prev (#72244546) Next (#72244548) →

#72244547

order10368
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C3

Aliases

No aliases for this group.

Splittings

FittingC32×Q8 ⋊ C3⋊GL(2,𝔽3)
derivedSL(2,𝔽3)×PSU(3,𝔽2)⋊C3 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC2#72213290
commutator, cosocle, hypocenterSL(2,𝔽3)×PSU(3,𝔽2)⋊C3C2
FittingC32×Q8C3⋊GL(2,𝔽3)
socleC2×C32A4⋊GL(2,𝔽3)
LCSG ▷ SL(2,𝔽3)×PSU(3,𝔽2)⋊C3
UCSC1 ◁ C2

745 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 745 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 17
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 17 points:

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

Polycyclic presentation:

2,3^2,2^6,3^2|d^2=i,e^2=i,f^2=h,g^2=h,b^a=b^2,c^a=c^2,d^a=ei,d^c=e,e^a=di,e^c=dei,e^d=ei,f^a=gh,f^b=g,g^a=fh,g^b=fgh,g^f=gh,j^a=jk,j^b=k,j^f=k,j^g=jk^2,j^h=j^2,k^a=k^2,k^b=j^2k^2,k^f=j^2,k^g=j^2k^2,k^h=k^2

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofGL(2,𝔽3)×AGL(2,𝔽3)
maximal normal subgroup ofGL(2,𝔽3)×AGL(2,𝔽3)

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