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#72340856

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4C2×#4626
commutator, hypocenter#95⋊C3C22×C4
FrattiniC22×C4C22×S4
FittingC2×#2397D3
socleC23C22×SL(2,ℤ/4)
prime radicalC2×#2397⋊C3C2
cosocle#392⋊C3C23
hypercenterC2×C4#4626
LCSG ▷ #95⋊C3
UCSC1 ◁ C4 ◁ C2×C4

882 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^4,3,2^5|b^2=d,d^2=h,f^2=i,g^2=ij,b^a=bij,e^a=ej,e^b=e^2,f^b=fg,f^c=fhi,f^d=fhi,f^e=ghi,g^b=ghj,g^c=ghij,g^d=ghij,g^e=fgh,g^f=gh,i^b=hj,i^e=ij,j^b=hi,j^e=hi

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340858, #72340860
maximal normal subgroup of#72340860

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