D₈₈₁₆ #72213699

order	17632
automorphism	Aut(G) order 35546112
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₈₁₆

Splittings

Fitting	C₁₆×C₁₉×C₂₉ ⋊ C₂
Hall	C₁₉×C₂₉ ⋊ D₁₆
Sylow	C₂₉ ⋊ D₃₀₄
Sylow	C₁₉ ⋊ D₄₆₄

Subgroups

	subgroup	G/H
center	C₂	D₄₄₀₈
commutator, cosocle	C₈×C₁₉×C₂₉	C₂²
Frattini	C₈	C₂×D₅₅₁
Fitting	C₁₆×C₁₉×C₂₉	C₂
socle	C₂×C₁₉×C₂₉	D₈
hypercenter	C₁₆	D₅₅₁
hypocenter	C₁₉×C₂₉	D₁₆
LCS	G ▷ C₈×C₁₉×C₂₉ ▷ C₄×C₁₉×C₂₉ ▷ C₂×C₁₉×C₂₉ ▷ C₁₉×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 8816

A minimal faithful permutation representation — 2 generators on 64 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2^5,19,29|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^18,g^a=g^28

Conjugacy classes & character table

4411 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₆₃₂, D₂₆₄₄₈
maximal normal subgroup of	D₁₇₆₃₂
maximal quotient of	Dic₈₈₁₆, D₁₇₆₃₂, D₂₆₄₄₈

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, 2, 19, 29]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
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.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8816 #72213699