D₁₇₄₆₄ #72273147

order	34928
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₇₄₆₄

Splittings

Fitting	C₈×C₃₇×C₅₉ ⋊ C₂
Hall	C₃₇×C₅₉ ⋊ D₈
Sylow	C₅₉ ⋊ D₂₉₆
Sylow	C₃₇ ⋊ D₄₇₂

Subgroups

	subgroup	G/H
center	C₂	D₈₇₃₂
commutator, cosocle	C₄×C₃₇×C₅₉	C₂²
Frattini	C₄	C₂×D₂₁₈₃
Fitting	C₈×C₃₇×C₅₉	C₂
socle	C₂×C₃₇×C₅₉	D₄
hypercenter	C₈	D₂₁₈₃
hypocenter	C₃₇×C₅₉	D₈
LCS	G ▷ C₄×C₃₇×C₅₉ ▷ C₂×C₃₇×C₅₉ ▷ C₃₇×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 104
transitive degree	μ_t(G) = 17464

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);(2,8)(3,7)(4,6)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(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:

2^4,37,59|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^36,f^a=f^58

Conjugacy classes & character table

8735 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, 2, 2, 2, 37, 59]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^36);;
gap> SetConjugate(coll, 6, 1, F.6^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D17464 #72273147