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#72245816

order2304
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2C22×#63694
commutator, cosocle#41⋊C9C24
FrattiniC3×C4C23×S4
FittingC3×H(7,𝔽2)D3
socleC2×C3C22×GL(2,ℤ/4)
hypercenter2−1+4(C22⋊C9)⋊C2
hypocenterQ8⋊C9C22×D4
LCSG ▷ #41⋊C9 ▷ Q8⋊C9
UCSC1 ◁ C2 ◁ #41 ◁ 2−1+4

3217 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 3217 conjugacy classes of subgroups — too many to draw. Only its 188 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) = 57
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 4 generators on 57 points:

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

Polycyclic presentation:

2^5,3^2,2^3|a^2=j,d^2=j,e^2=j,f^3=g,h^2=j,i^2=j,c^a=cj,d^b=de,e^b=ej,e^d=ej,f^b=f^2g^2,g^b=g^2,h^b=ij,h^f=hi,i^b=hj,i^f=h,i^h=ij

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245882, #72253040
maximal normal subgroup of#72253040

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