Finite Group Databasebeta

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#72225978

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

Aliases

No aliases for this group.

Splittings

FittingC22×H(5,𝔽2) ⋊ C3×D3

Subgroups

subgroupG/H
center, Frattini, socleC22C2×A4×S4
commutator, cosocleC2×#994C22×C3
FittingC22×H(5,𝔽2)C3×D3
hypercenterC23A4×S4
hypocenter#994C3×D4
LCSG ▷ C2×#994 ▷ #994
UCSC1 ◁ C22 ◁ C23

900 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 36
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 36 points:

(2,4)(5,21,8)(6,33,29)(7,35,12)(10,20,19)(11,26,31)(13,17,30)(14,32,23)(15,27,16)(18,36,25)(24,28,34);(1,2,3,4)(5,11,10,15)(6,18,14,7)(8,29,19,23)(9,36,22,35)(12,16,25,31)(13,32,28,33)(17,27,34,26)(20,30,21,24)

Polycyclic presentation:

2^2,3,2^3,3,2^3|a^2=d,e^2=j,f^2=j,h^2=j,i^2=j,b^a=bd,e^c=ef,f^c=e,f^e=fj,g^a=g^2,h^a=ij,h^g=i,i^a=hj,i^g=hij,i^h=ij

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226211, #72226221, #72226236
maximal normal subgroup of#72226211, #72226221, #72226236

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