Finite Group Databasebeta

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#72253045

order2304
automorphism?
homologyAbelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#62654 ⋊ C32

Subgroups

subgroupG/H
center, Frattini, socleC22C22×A42
commutator, hypocenterH(5,𝔽2)C2×C32×C4
Fitting#62654C32
prime radicalC2×(#41⋊C3)⋊A4C2
cosocleC2×H(5,𝔽2)C22×C32
hypercenterC4⋊C4A42
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C22 ◁ C4⋊C4

1138 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 1138 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 192

A minimal faithful permutation representation — 2 generators on 52 points:

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

Polycyclic presentation:

2^2,3^2,2^6|a^2=e,b^2=j,f^2=j,g^2=j,b^a=bj,f^c=gj,f^d=fgj,g^c=fg,g^d=fj,g^f=gj,h^c=fh,h^d=i,h^f=hj,i^c=fgij,i^d=hi,i^g=ij

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72253052
maximal normal subgroup of#72253052

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