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#72277623

order4374
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C33

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14555252 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC3C3⋊#92391
commutator, cosocle, hypocenterC32×H(3,𝔽3)C2×C32
Frattini, socleC32C32×C3⋊(C3⋊D3)
Fitting#14555252C2
LCSG ▷ C32×H(3,𝔽3)
UCSC1 ◁ C3

3757 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 3757 conjugacy classes of subgroups — too many to draw. Only its 181 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 — 4 generators on 54 points:

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

Polycyclic presentation:

2,3^7|c^b=ch,d^a=d^2,e^a=e^2,e^b=eg,e^d=eh,f^a=f^2,f^b=fg,f^c=fg,g^a=g^2

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72277725

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