Finite Group Databasebeta

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#72341767

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22C2×#1020⋊C2
commutator, cosocleC2×#997C23
FrattiniC24C22×S4
Fitting#32899D3
prime radical#32899⋊C3C2
hypercenterC2×C4#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ C2×#997 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4

1464 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 36
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341785, #72341839, #72341870
maximal normal subgroup of#72341785, #72341839, #72341870
maximal quotient of#72342072

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.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, 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, 3, 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, 3, 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);