D₈₃₄₀ #72213288

order	16680
automorphism	Aut(G) order 18414720
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₃₄₀

Splittings

Fitting	C₃×C₄×C₅×C₁₃₉ ⋊ C₂
Hall	C₃×C₅×C₁₃₉ ⋊ D₄
Hall	C₅×C₁₃₉ ⋊ D₁₂
Hall	C₃×C₁₃₉ ⋊ D₂₀
Sylow	C₁₃₉ ⋊ D₆₀
Hall	C₃×C₅ ⋊ D₅₅₆
Sylow	C₅ ⋊ D₁₆₆₈
Sylow	C₃ ⋊ D₂₇₈₀

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₀₈₅
commutator, socle, cosocle	C₂×C₃×C₅×C₁₃₉	C₂²
Fitting	C₃×C₄×C₅×C₁₃₉	C₂
hypercenter	C₄	D₂₀₈₅
hypocenter	C₃×C₅×C₁₃₉	D₄
LCS	G ▷ C₂×C₃×C₅×C₁₃₉ ▷ C₃×C₅×C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 151
transitive degree	μ_t(G) = 8340

A minimal faithful permutation representation — 2 generators on 151 points:

(1,2,3,4)(5,6,7)(8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151);(2,4)(6,7)(9,12)(10,11)(14,151)(15,150)(16,149)(17,148)(18,147)(19,146)(20,145)(21,144)(22,143)(23,142)(24,141)(25,140)(26,139)(27,138)(28,137)(29,136)(30,135)(31,134)(32,133)(33,132)(34,131)(35,130)(36,129)(37,128)(38,127)(39,126)(40,125)(41,124)(42,123)(43,122)(44,121)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)

Polycyclic presentation:

2^3,3,5,139|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^138

Conjugacy classes & character table

4173 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₆₆₈₀, D₂₅₀₂₀
maximal normal subgroup of	D₁₆₆₈₀
maximal quotient of	Dic₈₃₄₀, D₁₆₆₈₀, D₂₅₀₂₀

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, 3, 5, 139]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^138);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8340 #72213288