D₁₃₀₈ #72210170

order	2616
automorphism	Aut(G) order 565056
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₀₈

Splittings

Fitting	C₃×C₄×C₁₀₉ ⋊ C₂
Hall	C₃×C₁₀₉ ⋊ D₄
Sylow	C₁₀₉ ⋊ D₁₂
Sylow	C₃ ⋊ D₄₃₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₂₇
commutator, socle, cosocle	C₂×C₃×C₁₀₉	C₂²
Fitting	C₃×C₄×C₁₀₉	C₂
hypercenter	C₄	D₃₂₇
hypocenter	C₃×C₁₀₉	D₄
LCS	G ▷ C₂×C₃×C₁₀₉ ▷ C₃×C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 116
transitive degree	μ_t(G) = 1308

A minimal faithful permutation representation — 2 generators on 116 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);(2,4)(6,7)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)

Polycyclic presentation:

2^3,3,109|b^2=c,b^a=bc,d^a=d^2,e^a=e^108

Conjugacy classes & character table

657 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₆₁₆, D₃₉₂₄, D₆₅₄₀, …
maximal normal subgroup of	D₂₆₁₆
maximal quotient of	D₂₆₁₆, D₃₉₂₄, Dic₁₃₀₈, …

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, 3, 109]);;
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.5^108);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1308 #72210170