D₂₅₄₀ #72210461

order	5080
automorphism	Aut(G) order 2560320
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₅₄₀

Splittings

Fitting	C₄×C₅×C₁₂₇ ⋊ C₂
Hall	C₅×C₁₂₇ ⋊ D₄
Sylow	C₁₂₇ ⋊ D₂₀
Sylow	C₅ ⋊ D₅₀₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₃₅
commutator, socle, cosocle	C₂×C₅×C₁₂₇	C₂²
Fitting	C₄×C₅×C₁₂₇	C₂
hypercenter	C₄	D₆₃₅
hypocenter	C₅×C₁₂₇	D₄
LCS	G ▷ C₂×C₅×C₁₂₇ ▷ C₅×C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 136
transitive degree	μ_t(G) = 2540

A minimal faithful permutation representation — 2 generators on 136 points:

(1,2,3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136);(2,4)(6,9)(7,8)(11,136)(12,135)(13,134)(14,133)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,126)(22,125)(23,124)(24,123)(25,122)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)

Polycyclic presentation:

2^3,5,127|b^2=c,b^a=bc,d^a=d^4,e^a=e^126

Conjugacy classes & character table

1273 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₀₈₀, D₇₆₂₀, D₁₂₇₀₀, …
maximal normal subgroup of	D₅₀₈₀
maximal quotient of	D₅₀₈₀, Dic₂₅₄₀, D₇₆₂₀, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 127]);;
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.5^126);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D2540 #72210461