Finite Group Databasebeta

← Prev (#72228366) Next (#72228368) →

#72228367

order4608
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24×C3

Aliases

No aliases for this group.

Splittings

FittingC2×H(7,𝔽2) ⋊ C3⋊D3

Subgroups

subgroupG/H
center, Frattini, socleC22C22×A4⋊S4
commutator, cosocleC2×Ω+(4,𝔽3)C23
FittingC2×H(7,𝔽2)C3⋊D3
hypercenterC2×D4A4⋊S4
hypocenterΩ+(4,𝔽3)C2×D4
LCSG ▷ C2×Ω+(4,𝔽3) ▷ Ω+(4,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

5516 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 5516 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 52
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 52 points:

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

Polycyclic presentation:

2^4,3^2,2^5|g^2=k,h^2=k,b^a=bd,c^b=ck,e^a=e^2,f^a=f^2,g^a=gk,g^e=hk,g^f=ghk,h^a=ghk,h^e=gh,h^f=gk,h^g=hk,i^a=gi,i^e=gi,i^f=j,i^g=ik,j^a=hijk,j^e=ghjk,j^f=ij,j^h=jk

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

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