D₂₉₁₃₃ #72292737

order	58266
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₉₁₃₃

Splittings

Fitting, derived, Hall	C₁₃×C₂₇×C₈₃ ⋊ C₂
Hall	C₂₇×C₈₃ ⋊ D₁₃
Hall	C₁₃×C₈₃ ⋊ D₂₇
Hall	C₁₃×C₂₇ ⋊ D₈₃
Sylow	C₈₃ ⋊ D₃₅₁
Sylow	C₂₇ ⋊ D₁₀₇₉
Sylow	C₁₃ ⋊ D₂₂₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₃×C₂₇×C₈₃	C₂
Frattini	C₉	D₃₂₃₇
socle	C₃×C₁₃×C₈₃	D₉
LCS	G ▷ C₁₃×C₂₇×C₈₃
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 29133

A minimal faithful permutation representation — 2 generators on 123 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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(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)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(42,123)(43,122)(44,121)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)

Polycyclic presentation:

2,3,13,83,3^2|b^3=e,e^3=f,b^a=b^2e^2f^2,c^a=c^12,d^a=d^82,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

14568 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");;
gap> coll := SingleCollector(F, [2, 3, 13, 83, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^12);;
gap> SetConjugate(coll, 4, 1, F.4^82);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D29133 #72292737