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#72245693

order3456
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

derived#997⋊C9 ⋊ C22

Subgroups

subgroupG/H
centerC2C2×#11724027
commutator, cosocle, hypocenter#997⋊C9C22
Frattini, socleC2×C3C2×A4⋊S4
FittingC3×#585C3⋊D3
hypercenterC4#11724027
LCSG ▷ #997⋊C9
UCSC1 ◁ C2 ◁ C4

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

normal subgroups only

This group has 682 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) = 57
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 57 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245730, #72245882
maximal normal subgroup of#72245730, #72245882

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