D₃₀₀₂₀ #72295358

order	60040
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₀₂₀

Splittings

Fitting	C₄×C₅×C₁₉×C₇₉ ⋊ C₂
Hall	C₅×C₁₉×C₇₉ ⋊ D₄
Hall	C₁₉×C₇₉ ⋊ D₂₀
Hall	C₅×C₇₉ ⋊ D₇₆
Hall	C₅×C₁₉ ⋊ D₃₁₆
Sylow	C₇₉ ⋊ D₃₈₀
Sylow	C₁₉ ⋊ D₁₅₈₀
Sylow	C₅ ⋊ D₆₀₀₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₅₀₅
commutator, socle, cosocle	C₂×C₅×C₁₉×C₇₉	C₂²
Fitting	C₄×C₅×C₁₉×C₇₉	C₂
hypercenter	C₄	D₇₅₀₅
hypocenter	C₅×C₁₉×C₇₉	D₄
LCS	G ▷ C₂×C₅×C₁₉×C₇₉ ▷ C₅×C₁₉×C₇₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 30020

A minimal faithful permutation representation — 2 generators on 107 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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(2,4)(6,9)(7,8)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2^3,5,19,79|b^2=c,b^a=bc,d^a=d^4,e^a=e^18,f^a=f^78

Conjugacy classes & character table

15013 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, 19, 79]);;
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.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^78);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30020 #72295358