D₁₈₉₈₀ #72273802

order	37960
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₈₉₈₀

Splittings

Fitting	C₄×C₅×C₁₃×C₇₃ ⋊ C₂
Hall	C₅×C₁₃×C₇₃ ⋊ D₄
Hall	C₁₃×C₇₃ ⋊ D₂₀
Hall	C₅×C₇₃ ⋊ D₅₂
Sylow	C₇₃ ⋊ D₂₆₀
Hall	C₅×C₁₃ ⋊ D₂₉₂
Sylow	C₁₃ ⋊ D₁₄₆₀
Sylow	C₅ ⋊ D₃₇₉₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₄₇₄₅
commutator, socle, cosocle	C₂×C₅×C₁₃×C₇₃	C₂²
Fitting	C₄×C₅×C₁₃×C₇₃	C₂
hypercenter	C₄	D₄₇₄₅
hypocenter	C₅×C₁₃×C₇₃	D₄
LCS	G ▷ C₂×C₅×C₁₃×C₇₃ ▷ C₅×C₁₃×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 18980

A minimal faithful permutation representation — 2 generators on 95 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);(2,4)(6,9)(7,8)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2^3,5,13,73|b^2=c,b^a=bc,d^a=d^4,e^a=e^12,f^a=f^72

Conjugacy classes & character table

9493 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, 5, 13, 73]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D18980 #72273802