D₂₇₃₇₆ #72290684

order	54752
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₃₇₆

Splittings

Fitting	C₁₆×C₂₉×C₅₉ ⋊ C₂
Hall	C₂₉×C₅₉ ⋊ D₁₆
Sylow	C₅₉ ⋊ D₄₆₄
Sylow	C₂₉ ⋊ D₉₄₄

Subgroups

	subgroup	G/H
center	C₂	D₁₃₆₈₈
commutator, cosocle	C₈×C₂₉×C₅₉	C₂²
Frattini	C₈	C₂×D₁₇₁₁
Fitting	C₁₆×C₂₉×C₅₉	C₂
socle	C₂×C₂₉×C₅₉	D₈
hypercenter	C₁₆	D₁₇₁₁
hypocenter	C₂₉×C₅₉	D₁₆
LCS	G ▷ C₈×C₂₉×C₅₉ ▷ C₄×C₂₉×C₅₉ ▷ C₂×C₂₉×C₅₉ ▷ C₂₉×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 104
transitive degree	μ_t(G) = 27376

A minimal faithful permutation representation — 2 generators on 104 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2^5,29,59|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^28,g^a=g^58

Conjugacy classes & character table

13691 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, 2, 2, 2, 2, 29, 59]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> SetConjugate(coll, 7, 1, F.7^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27376 #72290684