Finite Group Databasebeta

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#72342005

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#73081
commutator, cosocleC4×#997C22
FrattiniC23×C4C2×S4
Fitting#32926D3
prime radicalC4⋊(B(4,𝔽2)⋊C3)C2
hypercenterC2×C4#1020⋊C2
hypocenter#997D8
LCSG ▷ C4×#997 ▷ C2×#997 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4

546 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 23 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) = 24
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 24 points:

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

Polycyclic presentation:

2^4,3,2^5|b^2=j,c^2=d,f^2=j,g^2=j,b^a=bc,c^a=cd,c^b=cd,e^b=e^2,f^a=fhj,f^b=fg,f^e=g,g^a=gij,g^b=gj,g^e=fgj,g^f=gj,h^b=hi,h^e=i,h^g=hj,i^e=hi,i^f=ij

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342017, #72342072, #72342082
maximal normal subgroup of#72342017, #72342072, #72342082

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