D₁₃₈₀ #72210193

order	2760
automorphism	Aut(G) order 485760
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₈₀

Splittings

Fitting	C₃×C₄×C₅×C₂₃ ⋊ C₂
Hall	C₃×C₅×C₂₃ ⋊ D₄
Hall	C₅×C₂₃ ⋊ D₁₂
Hall	C₃×C₂₃ ⋊ D₂₀
Sylow	C₂₃ ⋊ D₆₀
Hall	C₃×C₅ ⋊ D₉₂
Sylow	C₅ ⋊ D₂₇₆
Sylow	C₃ ⋊ D₄₆₀

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₄₅
commutator, socle, cosocle	C₂×C₃×C₅×C₂₃	C₂²
Fitting	C₃×C₄×C₅×C₂₃	C₂
hypercenter	C₄	D₃₄₅
hypocenter	C₃×C₅×C₂₃	D₄
LCS	G ▷ C₂×C₃×C₅×C₂₃ ▷ C₃×C₅×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 1380

A minimal faithful permutation representation — 2 generators on 35 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);(2,4)(6,7)(9,12)(10,11)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)

Polycyclic presentation:

2^3,3,5,23|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^22

Conjugacy classes & character table

693 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₇₆₀, D₄₁₄₀, D₆₉₀₀, …
maximal normal subgroup of	D₂₇₆₀
maximal quotient of	D₂₇₆₀, D₄₁₄₀, Dic₁₃₈₀, …

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, 23]);;
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.6^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1380 #72210193