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#72240030

order12288
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C22×C42

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72240028 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72210010
commutator, Fitting, cosocle, hypocenter#72240028C3
FrattiniC24×C42C22⋊A4
socleC23#72210009
LCSG ▷ #72240028
UCSC1 ◁ C2

16103 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 16103 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 48

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

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

Polycyclic presentation:

3,2^12|b^2=f,c^2=g,d^2=fghlm,e^2=fjklm,f^2=m,g^2=l,h^2=lm,i^2=m,j^2=m,b^a=cfklm,c^a=bck,c^b=ck,d^a=efghlm,d^b=di,d^c=dh,e^a=de,e^b=ehijm,e^c=ehi,f^a=gm,g^a=fgk,h^a=jm,h^b=hm,h^c=hlm,i^a=hilm,i^b=im,i^c=im,j^a=hjm,j^b=jlm,j^c=jl,l^a=lm,m^a=l

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240031

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", "l", "m");;
gap> coll := SingleCollector(F, [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.6*F.7*F.8*F.12*F.13);;
gap> SetPower(coll, 5, F.6*F.10*F.11*F.12*F.13);;
gap> SetPower(coll, 6, F.13);;
gap> SetPower(coll, 7, F.12);;
gap> SetPower(coll, 8, F.12*F.13);;
gap> SetPower(coll, 9, F.13);;
gap> SetPower(coll, 10, F.13);;
gap> SetConjugate(coll, 2, 1, F.3*F.6*F.11*F.12*F.13);;
gap> SetConjugate(coll, 3, 1, F.2*F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 4, 1, F.5*F.6*F.7*F.8*F.12*F.13);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10*F.13);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.7*F.13);;
gap> SetConjugate(coll, 7, 1, F.6*F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.10*F.13);;
gap> SetConjugate(coll, 8, 2, F.8*F.13);;
gap> SetConjugate(coll, 8, 3, F.8*F.12*F.13);;
gap> SetConjugate(coll, 9, 1, F.8*F.9*F.12*F.13);;
gap> SetConjugate(coll, 9, 2, F.9*F.13);;
gap> SetConjugate(coll, 9, 3, F.9*F.13);;
gap> SetConjugate(coll, 10, 1, F.8*F.10*F.13);;
gap> SetConjugate(coll, 10, 2, F.10*F.12*F.13);;
gap> SetConjugate(coll, 10, 3, F.10*F.12);;
gap> SetConjugate(coll, 12, 1, F.12*F.13);;
gap> SetConjugate(coll, 13, 1, F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);