Dic₅₉₈₆ #72224258

order	23944
automorphism	Aut(G) order 68958720
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₉₈₆

Splittings

Hall	C₄₁×C₇₃ ⋊ Q₈
Sylow	C₇₃ ⋊ Dic₈₂
Sylow	C₄₁ ⋊ Dic₁₄₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₉₉₃
commutator, socle, prime radical, cosocle	C₂×C₄₁×C₇₃	C₂²
Fitting	C₄×C₄₁×C₇₃	C₂
hypercenter	C₄	D₂₉₉₃
hypocenter	C₄₁×C₇₃	Q₈
LCS	G ▷ C₂×C₄₁×C₇₃ ▷ C₄₁×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 122
transitive degree	μ_t(G) = 23944

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

Polycyclic presentation:

2^3,41,73|a^2=c,b^2=c,b^a=bc,d^a=d^40,e^a=e^72

Conjugacy classes & character table

5989 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₉₇₂
maximal normal subgroup of	Dic₁₁₉₇₂

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, 41, 73]);;
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.4^40);;
gap> SetConjugate(coll, 5, 1, F.5^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5986 #72224258