D₁₇₆₄₀ #72273229

order	35280
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₇₆₄₀

Splittings

Fitting	C₅×C₈×C₉×C₄₉ ⋊ C₂
Hall	C₅×C₉×C₄₉ ⋊ D₈
Hall	C₉×C₄₉ ⋊ D₄₀
Hall	C₅×C₄₉ ⋊ D₇₂
Sylow	C₄₉ ⋊ D₃₆₀
Hall	C₅×C₉ ⋊ D₃₉₂
Sylow	C₉ ⋊ D₁₉₆₀
Sylow	C₅ ⋊ D₃₅₂₈

Subgroups

	subgroup	G/H
center	C₂	D₈₈₂₀
commutator, cosocle	C₄×C₅×C₉×C₄₉	C₂²
Frattini	C₃×C₄×C₇	C₂×D₁₀₅
Fitting	C₅×C₈×C₉×C₄₉	C₂
socle	C₂×C₃×C₅×C₇	D₈₄
hypercenter	C₈	D₂₂₀₅
hypocenter	C₅×C₉×C₄₉	D₈
LCS	G ▷ C₄×C₅×C₉×C₄₉ ▷ C₂×C₅×C₉×C₄₉ ▷ C₅×C₉×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

198 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 71
transitive degree	μ_t(G) = 17640

A minimal faithful permutation representation — 2 generators on 71 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);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,22)(20,21)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2^4,3,5,7,3,7|b^2=cd,c^2=d,e^3=h,g^7=i,b^a=bc,c^a=cd,e^a=e^2h^2,f^a=f^4,g^a=g^6i^6,h^a=h^2,i^a=i^6

Conjugacy classes & character table

8823 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, 3, 5, 7, 3, 7]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
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*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^6*F.9^6);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);