D₆₄₆₀ #72212218

order	12920
automorphism	Aut(G) order 14883840
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₆₄₆₀

Splittings

Fitting	C₄×C₅×C₁₇×C₁₉ ⋊ C₂
Hall	C₅×C₁₇×C₁₉ ⋊ D₄
Hall	C₁₇×C₁₉ ⋊ D₂₀
Hall	C₅×C₁₉ ⋊ D₆₈
Hall	C₅×C₁₇ ⋊ D₇₆
Sylow	C₁₉ ⋊ D₃₄₀
Sylow	C₁₇ ⋊ D₃₈₀
Sylow	C₅ ⋊ D₁₂₉₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₆₁₅
commutator, socle, cosocle	C₂×C₅×C₁₇×C₁₉	C₂²
Fitting	C₄×C₅×C₁₇×C₁₉	C₂
hypercenter	C₄	D₁₆₁₅
hypocenter	C₅×C₁₇×C₁₉	D₄
LCS	G ▷ C₂×C₅×C₁₇×C₁₉ ▷ C₅×C₁₇×C₁₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 6460

A minimal faithful permutation representation — 2 generators on 45 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);(2,4)(6,9)(7,8)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)

Polycyclic presentation:

2^3,5,17,19|b^2=c,b^a=bc,d^a=d^4,e^a=e^16,f^a=f^18

Conjugacy classes & character table

3233 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₂₉₂₀, D₁₉₃₈₀, D₃₂₃₀₀
maximal normal subgroup of	D₁₂₉₂₀
maximal quotient of	D₁₂₉₂₀, Dic₆₄₆₀, D₁₉₃₈₀, …

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, 17, 19]);;
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.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D6460 #72212218