Finite Group Databasebeta

← Prev (#72226084) Next (#72226086) →

#72226085

order2304
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

FittingC2×#585 ⋊ C3⋊D3

Subgroups

subgroupG/H
center, Frattini, socleC22C2×A4⋊S4
commutator, cosocleC2×Ω+(4,𝔽3)C22
FittingC2×#585C3⋊D3
hypercenterC2×C4A4⋊S4
hypocenterΩ+(4,𝔽3)D4
LCSG ▷ C2×Ω+(4,𝔽3) ▷ Ω+(4,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

929 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 929 conjugacy classes of subgroups — too many to draw. Only its 39 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:

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

Polycyclic presentation:

2^3,3^2,2^5|a^2=c,f^2=j,g^2=j,b^a=bcj,d^a=d^2,d^b=d^2,e^a=e^2,e^b=e^2,f^a=gj,f^b=gj,f^d=g,f^e=g,g^a=fj,g^b=fj,g^d=fgj,g^e=fgj,g^f=gj,h^a=fhi,h^b=fhi,h^d=i,h^e=fgh,h^g=hj,i^a=fgij,i^b=fgij,i^d=hi,i^e=fij,i^f=ij

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226335, #72226339, #72228367
maximal normal subgroup of#72226335, #72226339, #72228367

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