Dic₁₄₅₁₄ #72291260

order	58056
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₅₁₄

Splittings

Hall	C₃×C₄₁×C₅₉ ⋊ Q₈
Hall	C₄₁×C₅₉ ⋊ Dic₆
Hall	C₃×C₅₉ ⋊ Dic₈₂
Hall	C₃×C₄₁ ⋊ Dic₁₁₈
Sylow	C₅₉ ⋊ Dic₂₄₆
Sylow	C₄₁ ⋊ Dic₃₅₄
Sylow	C₃ ⋊ Dic₄₈₃₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₂₅₇
commutator, socle, prime radical, cosocle	C₂×C₃×C₄₁×C₅₉	C₂²
Fitting	C₃×C₄×C₄₁×C₅₉	C₂
hypercenter	C₄	D₇₂₅₇
hypocenter	C₃×C₄₁×C₅₉	Q₈
LCS	G ▷ C₂×C₃×C₄₁×C₅₉ ▷ C₃×C₄₁×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 58056

A minimal faithful permutation representation — 2 generators on 111 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);(1,5,3,7)(2,6,4,8)(10,11)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)

Polycyclic presentation:

2^3,3,41,59|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^40,f^a=f^58

Conjugacy classes & character table

14517 conjugacy classes. The character table is too large to display.

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 41, 59]);;
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.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^40);;
gap> SetConjugate(coll, 6, 1, F.6^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14514 #72291260