D₂₇₀₈₄ #72290416

order	54168
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₀₈₄

Splittings

Fitting	C₃×C₄×C₃₇×C₆₁ ⋊ C₂
Hall	C₃×C₃₇×C₆₁ ⋊ D₄
Hall	C₃₇×C₆₁ ⋊ D₁₂
Hall	C₃×C₆₁ ⋊ D₁₄₈
Hall	C₃×C₃₇ ⋊ D₂₄₄
Sylow	C₆₁ ⋊ D₄₄₄
Sylow	C₃₇ ⋊ D₇₃₂
Sylow	C₃ ⋊ D₉₀₂₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₇₇₁
commutator, socle, cosocle	C₂×C₃×C₃₇×C₆₁	C₂²
Fitting	C₃×C₄×C₃₇×C₆₁	C₂
hypercenter	C₄	D₆₇₇₁
hypocenter	C₃×C₃₇×C₆₁	D₄
LCS	G ▷ C₂×C₃×C₃₇×C₆₁ ▷ C₃×C₃₇×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 105
transitive degree	μ_t(G) = 27084

A minimal faithful permutation representation — 2 generators on 105 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);(2,4)(6,7)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(46,105)(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:

2^3,3,37,61|b^2=c,b^a=bc,d^a=d^2,e^a=e^36,f^a=f^60

Conjugacy classes & character table

13545 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, 3, 37, 61]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^36);;
gap> SetConjugate(coll, 6, 1, F.6^60);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27084 #72290416