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#72268997

order20736
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C24×C3

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2C22×#72213207
commutator, cosocle, hypocenter#997⋊C9C23×C3
Frattini, socleC2×C3C22×C3×A4⋊S4
FittingC32×H(7,𝔽2)C3⋊D3
hypercenterD4#72213207
LCSG ▷ #997⋊C9
UCSC1 ◁ C2 ◁ D4

7246 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 7246 conjugacy classes of subgroups — too many to draw. Only its 170 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 — 3 generators on 57 points:

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

Polycyclic presentation:

2^3,3^4,2^5|f^3=g,h^2=l,i^2=l,c^a=cl,e^b=e^2,f^b=f^2g^2,f^d=fg,g^b=g^2,h^b=hl,h^e=i,h^f=i,i^b=hi,i^e=hil,i^f=hil,i^h=il,j^b=ijkl,j^e=k,j^f=hij,j^i=jl,k^b=hkl,k^e=jk,k^f=hkl,k^h=kl

Conjugacy classes & character table

221 conjugacy classes — view the full character table.

Relationships

holomorph ofC2×Q8⋊C9, #41⋊C9

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 8, F.12);;
gap> SetPower(coll, 9, F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.12);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6^2*F.7^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 2, F.8*F.12);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.9);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.8*F.9*F.12);;
gap> SetConjugate(coll, 9, 6, F.8*F.9*F.12);;
gap> SetConjugate(coll, 9, 8, F.9*F.12);;
gap> SetConjugate(coll, 10, 2, F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 10, 5, F.11);;
gap> SetConjugate(coll, 10, 6, F.8*F.9*F.10);;
gap> SetConjugate(coll, 10, 9, F.10*F.12);;
gap> SetConjugate(coll, 11, 2, F.8*F.11*F.12);;
gap> SetConjugate(coll, 11, 5, F.10*F.11);;
gap> SetConjugate(coll, 11, 6, F.8*F.11*F.12);;
gap> SetConjugate(coll, 11, 8, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);