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D27027 #72290360

order54054
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD27027

Splittings

Fitting, derived, HallC7×C11×C13×C27 ⋊ C2
HallC11×C13×C27 ⋊ D7
HallC7×C13×C27 ⋊ D11
HallC7×C11×C27 ⋊ D13
HallC7×C11×C13 ⋊ D27
HallC13×C27 ⋊ D77
HallC11×C27 ⋊ D91
HallC7×C27 ⋊ D143
HallC11×C13 ⋊ D189
HallC7×C13 ⋊ D297
HallC7×C11 ⋊ D351
SylowC27 ⋊ D1001
SylowC13 ⋊ D2079
SylowC11 ⋊ D2457
SylowC7 ⋊ D3861

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC7×C11×C13×C27C2
FrattiniC9D3003
socleC3×C7×C11×C13D9
LCSG ▷ C7×C11×C13×C27
UCSC1

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 58
transitive degreeμt(G) = 27027

A minimal faithful permutation representation — 2 generators on 58 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34)(35,36,37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54,55,56,57,58);(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,34)(30,33)(31,32)(36,45)(37,44)(38,43)(39,42)(40,41)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2,3,7,11,13,3^2|b^3=f,f^3=g,b^a=b^2f^2g^2,c^a=c^6,d^a=d^10,e^a=e^12,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

13515 conjugacy classes. The character table is too large to display.

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");;
gap> coll := SingleCollector(F, [2, 3, 7, 11, 13, 3, 3]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);