D₈₆₈₄ #72213610

order	17368
automorphism	Aut(G) order 34597056
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₆₈₄

Splittings

Fitting	C₄×C₁₃×C₁₆₇ ⋊ C₂
Hall	C₁₃×C₁₆₇ ⋊ D₄
Sylow	C₁₆₇ ⋊ D₅₂
Sylow	C₁₃ ⋊ D₆₆₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₁₇₁
commutator, socle, cosocle	C₂×C₁₃×C₁₆₇	C₂²
Fitting	C₄×C₁₃×C₁₆₇	C₂
hypercenter	C₄	D₂₁₇₁
hypocenter	C₁₃×C₁₆₇	D₄
LCS	G ▷ C₂×C₁₃×C₁₆₇ ▷ C₁₃×C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 184
transitive degree	μ_t(G) = 8684

A minimal faithful permutation representation — 2 generators on 184 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184);(2,4)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(19,184)(20,183)(21,182)(22,181)(23,180)(24,179)(25,178)(26,177)(27,176)(28,175)(29,174)(30,173)(31,172)(32,171)(33,170)(34,169)(35,168)(36,167)(37,166)(38,165)(39,164)(40,163)(41,162)(42,161)(43,160)(44,159)(45,158)(46,157)(47,156)(48,155)(49,154)(50,153)(51,152)(52,151)(53,150)(54,149)(55,148)(56,147)(57,146)(58,145)(59,144)(60,143)(61,142)(62,141)(63,140)(64,139)(65,138)(66,137)(67,136)(68,135)(69,134)(70,133)(71,132)(72,131)(73,130)(74,129)(75,128)(76,127)(77,126)(78,125)(79,124)(80,123)(81,122)(82,121)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)

Polycyclic presentation:

2^3,13,167|b^2=c,b^a=bc,d^a=d^12,e^a=e^166

Conjugacy classes & character table

4345 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₃₆₈, D₂₆₀₅₂
maximal normal subgroup of	D₁₇₃₆₈
maximal quotient of	Dic₈₆₈₄, D₁₇₃₆₈, D₂₆₀₅₂

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, 13, 167]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^166);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8684 #72213610