D₃₁₈₆₀ #72301787

order	63720
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₁₈₆₀

Splittings

Fitting	C₄×C₅×C₂₇×C₅₉ ⋊ C₂
Hall	C₅×C₂₇×C₅₉ ⋊ D₄
Hall	C₂₇×C₅₉ ⋊ D₂₀
Hall	C₅×C₅₉ ⋊ D₁₀₈
Hall	C₅×C₂₇ ⋊ D₂₃₆
Sylow	C₅₉ ⋊ D₅₄₀
Sylow	C₂₇ ⋊ D₁₁₈₀
Sylow	C₅ ⋊ D₆₃₇₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₉₆₅
commutator, cosocle	C₂×C₅×C₂₇×C₅₉	C₂²
Frattini	C₂×C₉	C₂×D₈₈₅
Fitting	C₄×C₅×C₂₇×C₅₉	C₂
socle	C₂×C₃×C₅×C₅₉	C₂×D₉
hypercenter	C₄	D₇₉₆₅
hypocenter	C₅×C₂₇×C₅₉	D₄
LCS	G ▷ C₂×C₅×C₂₇×C₅₉ ▷ C₅×C₂₇×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 31860

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)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,36)(34,35)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Polycyclic presentation:

2^3,3,5,59,3^2|b^2=c,d^3=g,g^3=h,b^a=bc,d^a=d^2g^2h^2,e^a=e^4,f^a=f^58,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

15933 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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 59, 3, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^58);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31860 #72301787