D₈₈₂₀ #72213698

order	17640
automorphism	Aut(G) order 17781120
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₈₂₀

Splittings

Fitting	C₄×C₅×C₉×C₄₉ ⋊ C₂
Hall	C₅×C₉×C₄₉ ⋊ D₄
Hall	C₉×C₄₉ ⋊ D₂₀
Hall	C₅×C₄₉ ⋊ D₃₆
Sylow	C₄₉ ⋊ D₁₈₀
Hall	C₅×C₉ ⋊ D₁₉₆
Sylow	C₉ ⋊ D₉₈₀
Sylow	C₅ ⋊ D₁₇₆₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₂₀₅
commutator, cosocle	C₂×C₅×C₉×C₄₉	C₂²
Frattini	C₂×C₃×C₇	C₂×D₁₀₅
Fitting	C₄×C₅×C₉×C₄₉	C₂
socle	C₂×C₃×C₅×C₇	C₂×D₂₁
hypercenter	C₄	D₂₂₀₅
hypocenter	C₅×C₉×C₄₉	D₄
LCS	G ▷ C₂×C₅×C₉×C₄₉ ▷ C₅×C₉×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄

144 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 67
transitive degree	μ_t(G) = 8820

A minimal faithful permutation representation — 2 generators on 67 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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,18)(16,17)(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:

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

Conjugacy classes & character table

4413 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₆₄₀, D₂₆₄₆₀
maximal normal subgroup of	D₁₇₆₄₀
maximal quotient of	Dic₈₈₂₀, D₁₇₆₄₀, D₂₆₄₆₀

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