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#72228602

order3888
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, SylowH(5,𝔽3) ⋊ C2×D4

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle#92392C23
Frattini, socleC3(#157⋊D3)⋊D3
Fitting, hypocenterH(5,𝔽3)C2×D4
LCSG ▷ #92392 ▷ H(5,𝔽3)
UCSC1

727 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 727 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 54
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 54 points:

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

Polycyclic presentation:

2^4,3^5|a^2=d,b^a=bd,e^a=ef,e^b=e^2,e^c=e^2i,e^d=e^2i,f^a=ef^2,f^b=e^2fi^2,f^c=f^2i,f^d=f^2i,f^e=fi,g^a=fgi^2,g^b=g^2,g^c=g^2,g^d=egi,g^f=gi^2,h^a=ef^2hi^2,h^c=h^2,h^d=efhi,h^e=hi,h^f=hi^2,i^b=i^2

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228607
maximal normal subgroup of#72228607

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