Finite Group Databasebeta

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#72342825

order1536
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

FittingC22×M64 ⋊ D3

Subgroups

subgroupG/H
centerC2#11711598
commutatorA4×C16C2×C4
FrattiniC2×C16C2×S4
FittingC22×M64D3
socleC23#4509
prime radicalC2×#73035C2
cosocleC2×A4×C16C22
hypercenterM64S4
hypocenterA4#1419
LCSG ▷ A4×C16 ▷ C8×A4 ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ M64

346 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 96

A minimal faithful permutation representation — 2 generators on 36 points:

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

Polycyclic presentation:

2^7,3,2^2|b^2=d,c^2=ef,d^2=g,e^2=fg,f^2=g,b^a=bc,c^a=ce,c^b=ceg,d^a=dg,e^a=ef,e^b=ef,f^a=fg,f^b=fg,h^a=h^2,h^b=h^2,i^a=ij,i^b=ij,i^h=j,j^h=ij

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342827, S4×#1419, #72342908
maximal normal subgroup of#72342827, S4×#1419, #72342908

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