D₁₃₅₈₀ #72235517

order	27160
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₅₈₀

Splittings

Fitting	C₄×C₅×C₇×C₉₇ ⋊ C₂
Hall	C₅×C₇×C₉₇ ⋊ D₄
Hall	C₇×C₉₇ ⋊ D₂₀
Hall	C₅×C₉₇ ⋊ D₂₈
Sylow	C₉₇ ⋊ D₁₄₀
Hall	C₅×C₇ ⋊ D₃₈₈
Sylow	C₇ ⋊ D₁₉₄₀
Sylow	C₅ ⋊ D₂₇₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₃₉₅
commutator, socle, cosocle	C₂×C₅×C₇×C₉₇	C₂²
Fitting	C₄×C₅×C₇×C₉₇	C₂
hypercenter	C₄	D₃₃₉₅
hypocenter	C₅×C₇×C₉₇	D₄
LCS	G ▷ C₂×C₅×C₇×C₉₇ ▷ C₅×C₇×C₉₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 13580

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

Polycyclic presentation:

2^3,5,7,97|b^2=c,b^a=bc,d^a=d^4,e^a=e^6,f^a=f^96

Conjugacy classes & character table

6793 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₇₁₆₀
maximal normal subgroup of	D₂₇₁₆₀
maximal quotient of	Dic₁₃₅₈₀, D₂₇₁₆₀

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, 97]);;
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.6^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D13580 #72235517