D₁₄₉₈₀ #72236016

order	29960
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₉₈₀

Splittings

Fitting	C₄×C₅×C₇×C₁₀₇ ⋊ C₂
Hall	C₅×C₇×C₁₀₇ ⋊ D₄
Hall	C₇×C₁₀₇ ⋊ D₂₀
Hall	C₅×C₁₀₇ ⋊ D₂₈
Sylow	C₁₀₇ ⋊ D₁₄₀
Hall	C₅×C₇ ⋊ D₄₂₈
Sylow	C₇ ⋊ D₂₁₄₀
Sylow	C₅ ⋊ D₂₉₉₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₇₄₅
commutator, socle, cosocle	C₂×C₅×C₇×C₁₀₇	C₂²
Fitting	C₄×C₅×C₇×C₁₀₇	C₂
hypercenter	C₄	D₃₇₄₅
hypocenter	C₅×C₇×C₁₀₇	D₄
LCS	G ▷ C₂×C₅×C₇×C₁₀₇ ▷ C₅×C₇×C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 14980

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

Polycyclic presentation:

2^3,5,7,107|b^2=c,b^a=bc,d^a=d^4,e^a=e^6,f^a=f^106

Conjugacy classes & character table

7493 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₉₉₆₀
maximal normal subgroup of	D₂₉₉₆₀
maximal quotient of	Dic₁₄₉₈₀, D₂₉₉₆₀

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, 7, 107]);;
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.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^106);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D14980 #72236016