D₃₀₉₆₀ #72297780

order	61920
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₉₆₀

Splittings

Fitting	C₅×C₉×C₁₆×C₄₃ ⋊ C₂
Hall	C₅×C₉×C₄₃ ⋊ D₁₆
Hall	C₉×C₄₃ ⋊ D₈₀
Hall	C₅×C₄₃ ⋊ D₁₄₄
Hall	C₅×C₉ ⋊ D₆₈₈
Sylow	C₄₃ ⋊ D₇₂₀
Sylow	C₉ ⋊ D₃₄₄₀
Sylow	C₅ ⋊ D₆₁₉₂

Subgroups

	subgroup	G/H
center	C₂	D₁₅₄₈₀
commutator, cosocle	C₅×C₈×C₉×C₄₃	C₂²
Frattini	C₃×C₈	C₂×D₆₄₅
Fitting	C₅×C₉×C₁₆×C₄₃	C₂
socle	C₂×C₃×C₅×C₄₃	D₂₄
hypercenter	C₁₆	D₁₉₃₅
hypocenter	C₅×C₉×C₄₃	D₁₆
LCS	G ▷ C₅×C₈×C₉×C₄₃ ▷ C₄×C₅×C₉×C₄₃ ▷ C₂×C₅×C₉×C₄₃ ▷ C₅×C₉×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

168 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 30960

A minimal faithful permutation representation — 2 generators on 73 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,25)(19,24)(20,23)(21,22)(27,30)(28,29)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2^5,3,5,43,3|b^2=cd,c^2=de,d^2=e,f^3=i,b^a=bc,c^a=cd,d^a=de,f^a=f^2i^2,g^a=g^4,h^a=h^42,i^a=i^2

Conjugacy classes & character table

15483 conjugacy classes. The character table is too large to display.

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", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 5, 43, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^42);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);