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#72243787

order8748
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14555438 ⋊ C22
derived#10624932 ⋊ C22×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#10624932C22×C3
FrattiniC32C3×C3⋊#1113
Fitting#14555438C22
socleC3C3×#11758090
LCSG ▷ #10624932
UCSC1

3162 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 3162 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 81
transitive degreeμt(G) = 81

A minimal faithful permutation representation — 3 generators on 81 points:

(1,13,41,14,16,49,6,18,19,21,4,39,5,38,15,3,20,47)(2,76,33,64,17,80,11,50,44,22,31,74,10,60,48,51,9,66)(7,63,30,8,23,69,24,32,65,29,43,62,34,27,42,28,52,70)(12,71,45,40,46,72,36,35,37)(25,56,67,58,68,81,54,61,77,57,53,75,55,79,59,26,78,73);(2,24,11,7,10,34)(3,12)(4,15)(5,6)(8,54,29,25,28,55)(9,65,17,30,31,42)(13,37)(14,40)(16,41)(18,45)(19,20)(21,36)(22,57,64,26,51,58)(23,33,52,44,43,48)(27,67,63,59,32,77)(35,47)(38,72)(39,46)(49,71)(50,75,76,73,60,81)(53,62,68,70,78,69)(56,80,61,66,79,74);(1,2,7)(3,8,22)(4,17,43)(5,10,34)(6,11,24)(9,23,16)(12,26,25)(13,32,50)(14,29,51)(15,33,42)(18,27,60)(19,48,65)(20,31,52)(21,28,64)(30,41,44)(35,56,68)(36,58,54)(37,73,67)(38,63,76)(39,70,74)(40,57,55)(45,75,59)(46,61,53)(47,69,66)(49,62,80)(71,79,78)(72,81,77)

Polycyclic presentation:

2^2,3^7|f^3=i^2,d^a=d^2,d^b=d^2,e^a=e^2,e^b=e^2,f^a=f^2i,f^c=fi^2,f^d=fh,f^e=fh^2i,g^b=g^2,g^d=gi,h^a=hi^2,h^b=h^2i,h^d=hi,h^e=hi^2,i^a=i^2

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243793, #72244007, #72244066
maximal normal subgroup of#72244066

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