D₆₃₆₀ #72212178

order	12720
automorphism	Aut(G) order 10583040
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₆₃₆₀

Splittings

Fitting	C₃×C₅×C₈×C₅₃ ⋊ C₂
Hall	C₃×C₅×C₅₃ ⋊ D₈
Hall	C₅×C₅₃ ⋊ D₂₄
Hall	C₃×C₅₃ ⋊ D₄₀
Sylow	C₅₃ ⋊ D₁₂₀
Hall	C₃×C₅ ⋊ D₄₂₄
Sylow	C₅ ⋊ D₁₂₇₂
Sylow	C₃ ⋊ D₂₁₂₀

Subgroups

	subgroup	G/H
center	C₂	D₃₁₈₀
commutator, cosocle	C₃×C₄×C₅×C₅₃	C₂²
Frattini	C₄	C₂×D₇₉₅
Fitting	C₃×C₅×C₈×C₅₃	C₂
socle	C₂×C₃×C₅×C₅₃	D₄
hypercenter	C₈	D₇₉₅
hypocenter	C₃×C₅×C₅₃	D₈
LCS	G ▷ C₃×C₄×C₅×C₅₃ ▷ C₂×C₃×C₅×C₅₃ ▷ C₃×C₅×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 6360

A minimal faithful permutation representation — 2 generators on 69 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);(2,8)(3,7)(4,6)(10,11)(13,16)(14,15)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

3183 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₂₇₂₀, D₁₉₀₈₀, D₃₁₈₀₀
maximal normal subgroup of	D₁₂₇₂₀
maximal quotient of	D₁₂₇₂₀, Dic₆₃₆₀, D₁₉₀₈₀, …

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 53]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D6360 #72212178