Finite Group Databasebeta

← Prev (#72342720) Next (#72342722) →

#72342721

order1536
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#20612 ⋊ D3

Subgroups

subgroupG/H
centerC4C2×#1020⋊C2
commutator, hypocenter#997C22×C4
FrattiniC22×C4C22×S4
Fitting#20612D3
socleC2C2×#1020⋊C4
prime radical#11715827C2
cosocle#5950C23
hypercenterC2×C4#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C4 ◁ C2×C4

1389 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 1389 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,23,5,6,7,25,22,26)(2,38,11,32,9,36,31,12)(3,46,16,24,14,44,21,17)(4,48,18,19,20,50,43,45)(8,54,28,29,13,56,35,37)(10,60,33,55,30,58,53,34)(15,62,41,49,40,61,47,42)(27,63,51,52,39,64,57,59);(1,3,4,15,7,14,20,40)(2,8,10,27,9,13,30,39)(5,24,50,47,26,46,43,42,25,21,19,61,22,17,48,41,6,44,18,49,23,16,45,62)(11,37,60,57,32,56,53,52,38,35,34,63,31,29,58,51,12,54,33,59,36,28,55,64);(1,2)(3,13)(4,10)(5,11)(6,12)(7,9)(8,14)(15,39)(16,35)(17,37)(18,33)(19,34)(20,30)(21,28)(22,31)(23,36)(24,29)(25,38)(26,32)(27,40)(41,57)(42,59)(43,53)(44,56)(45,55)(46,54)(47,51)(48,58)(49,52)(50,60)(61,64)(62,63)

Polycyclic presentation:

2^4,3,2^5|b^2=d,d^2=j,f^2=j,g^2=j,b^a=bj,c^a=cj,c^b=cj,e^c=e^2,f^b=fij,f^c=fj,f^e=g,g^b=ghj,g^c=fg,g^e=fgj,g^f=gj,h^c=hi,h^e=hi,h^f=hj,i^e=h,i^g=ij

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342728, #72342778, #72342800, …
maximal normal subgroup of#72342728, #72342778, #72342800

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