D₈₂₈₄ #72213222

order	16568
automorphism	Aut(G) order 32208192
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₂₈₄

Splittings

Fitting	C₄×C₁₉×C₁₀₉ ⋊ C₂
Hall	C₁₉×C₁₀₉ ⋊ D₄
Sylow	C₁₀₉ ⋊ D₇₆
Sylow	C₁₉ ⋊ D₄₃₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₀₇₁
commutator, socle, cosocle	C₂×C₁₉×C₁₀₉	C₂²
Fitting	C₄×C₁₉×C₁₀₉	C₂
hypercenter	C₄	D₂₀₇₁
hypocenter	C₁₉×C₁₀₉	D₄
LCS	G ▷ C₂×C₁₉×C₁₀₉ ▷ C₁₉×C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 8284

A minimal faithful permutation representation — 2 generators on 132 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132);(2,4)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(25,132)(26,131)(27,130)(28,129)(29,128)(30,127)(31,126)(32,125)(33,124)(34,123)(35,122)(36,121)(37,120)(38,119)(39,118)(40,117)(41,116)(42,115)(43,114)(44,113)(45,112)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)

Polycyclic presentation:

2^3,19,109|b^2=c,b^a=bc,d^a=d^18,e^a=e^108

Conjugacy classes & character table

4145 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₆₅₆₈, D₂₄₈₅₂
maximal normal subgroup of	D₁₆₅₆₈
maximal quotient of	Dic₈₂₈₄, D₁₆₅₆₈, D₂₄₈₅₂

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 19, 109]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^18);;
gap> SetConjugate(coll, 5, 1, F.5^108);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8284 #72213222