D₈₆₃₆ #72213541

order	17272
automorphism	Aut(G) order 34820352
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₆₃₆

Splittings

Fitting	C₄×C₁₇×C₁₂₇ ⋊ C₂
Hall	C₁₇×C₁₂₇ ⋊ D₄
Sylow	C₁₂₇ ⋊ D₆₈
Sylow	C₁₇ ⋊ D₅₀₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₁₅₉
commutator, socle, cosocle	C₂×C₁₇×C₁₂₇	C₂²
Fitting	C₄×C₁₇×C₁₂₇	C₂
hypercenter	C₄	D₂₁₅₉
hypocenter	C₁₇×C₁₂₇	D₄
LCS	G ▷ C₂×C₁₇×C₁₂₇ ▷ C₁₇×C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 148
transitive degree	μ_t(G) = 8636

A minimal faithful permutation representation — 2 generators on 148 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148);(2,4)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(23,148)(24,147)(25,146)(26,145)(27,144)(28,143)(29,142)(30,141)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,130)(42,129)(43,128)(44,127)(45,126)(46,125)(47,124)(48,123)(49,122)(50,121)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)

Polycyclic presentation:

2^3,17,127|b^2=c,b^a=bc,d^a=d^16,e^a=e^126

Conjugacy classes & character table

4321 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₂₇₂, D₂₅₉₀₈
maximal normal subgroup of	D₁₇₂₇₂
maximal quotient of	Dic₈₆₃₆, D₁₇₂₇₂, D₂₅₉₀₈

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, 17, 127]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^126);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8636 #72213541