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#72272029

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, socleC22C22×#1020
commutator, cosocle, hypocenterQ8⋊SL(2,𝔽3)C23
Fitting#412D3
prime radical#11710697C2
hypercenterC42#1020
LCSG ▷ Q8⋊SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42

1952 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 1952 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 48
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 3 generators on 48 points:

(1,3,5,6,13,12)(2,8,10,7,14,16)(4,11)(9,15)(17,19,22,30)(18,23,26,27)(20,29,32,44,21,45,33,46,35,28,36,43)(24,38,39,41,25,48,34,42,40,37,31,47);(1,14)(2,3)(4,16)(5,9)(6,8)(7,13)(10,11)(12,15)(17,33,22,20)(18,34,26,24)(19,44,30,28)(21,32,36,35)(23,41,27,37)(25,39,31,40)(29,43,46,45)(38,47,42,48);(2,7)(3,11)(4,13)(5,12)(8,9)(14,15)(17,18,22,26)(19,27,30,23)(20,31,33,25)(21,34,36,24)(28,38,44,42)(29,37,46,41)(32,40,35,39)(43,47,45,48)

Polycyclic presentation:

2^3,3,2^6|a^2=ij,b^2=i,c^2=ij,e^2=j,f^2=j,g^2=i,h^2=i,b^a=bi,c^a=cij,d^a=d^2,e^a=fj,e^d=f,f^a=ej,f^d=efj,f^e=fj,g^a=gi,g^d=h,g^e=gi,g^f=gi,h^a=gh,h^d=ghi,h^f=hi,h^g=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72272057, #72272087, #72272107
maximal normal subgroup of#72272057, #72272087, #72272107

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