Dic₆₅₈₆ #72236166

order	26344
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₅₈₆

Splittings

Hall	C₃₇×C₈₉ ⋊ Q₈
Sylow	C₈₉ ⋊ Dic₇₄
Sylow	C₃₇ ⋊ Dic₁₇₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₂₉₃
commutator, socle, prime radical, cosocle	C₂×C₃₇×C₈₉	C₂²
Fitting	C₄×C₃₇×C₈₉	C₂
hypercenter	C₄	D₃₂₉₃
hypocenter	C₃₇×C₈₉	Q₈
LCS	G ▷ C₂×C₃₇×C₈₉ ▷ C₃₇×C₈₉
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 134
transitive degree	μ_t(G) = 26344

A minimal faithful permutation representation — 2 generators on 134 points:

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

Polycyclic presentation:

2^3,37,89|a^2=c,b^2=c,b^a=bc,d^a=d^36,e^a=e^88

Conjugacy classes & character table

6589 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₁₇₂
maximal normal subgroup of	Dic₁₃₁₇₂

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, 37, 89]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^36);;
gap> SetConjugate(coll, 5, 1, F.5^88);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6586 #72236166