D₁₃₇₈₀ #72235596

order	27560
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₇₈₀

Splittings

Fitting	C₄×C₅×C₁₃×C₅₃ ⋊ C₂
Hall	C₅×C₁₃×C₅₃ ⋊ D₄
Hall	C₁₃×C₅₃ ⋊ D₂₀
Hall	C₅×C₅₃ ⋊ D₅₂
Hall	C₅×C₁₃ ⋊ D₂₁₂
Sylow	C₅₃ ⋊ D₂₆₀
Sylow	C₁₃ ⋊ D₁₀₆₀
Sylow	C₅ ⋊ D₂₇₅₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₄₄₅
commutator, socle, cosocle	C₂×C₅×C₁₃×C₅₃	C₂²
Fitting	C₄×C₅×C₁₃×C₅₃	C₂
hypercenter	C₄	D₃₄₄₅
hypocenter	C₅×C₁₃×C₅₃	D₄
LCS	G ▷ C₂×C₅×C₁₃×C₅₃ ▷ C₅×C₁₃×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 13780

A minimal faithful permutation representation — 2 generators on 75 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);(2,4)(6,9)(7,8)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2^3,5,13,53|b^2=c,b^a=bc,d^a=d^4,e^a=e^12,f^a=f^52

Conjugacy classes & character table

6893 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₇₅₆₀
maximal normal subgroup of	D₂₇₅₆₀
maximal quotient of	Dic₁₃₇₈₀, D₂₇₅₆₀

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, 13, 53]);;
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.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D13780 #72235596