Dic₆₂₅₄ #72236043

order	25016
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₂₅₄

Splittings

Hall	C₅₃×C₅₉ ⋊ Q₈
Sylow	C₅₉ ⋊ Dic₁₀₆
Sylow	C₅₃ ⋊ Dic₁₁₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₁₂₇
commutator, socle, prime radical, cosocle	C₂×C₅₃×C₅₉	C₂²
Fitting	C₄×C₅₃×C₅₉	C₂
hypercenter	C₄	D₃₁₂₇
hypocenter	C₅₃×C₅₉	Q₈
LCS	G ▷ C₂×C₅₃×C₅₉ ▷ C₅₃×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 120
transitive degree	μ_t(G) = 25016

A minimal faithful permutation representation — 2 generators on 120 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);(1,5,3,7)(2,6,4,8)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)

Polycyclic presentation:

2^3,53,59|a^2=c,b^2=c,b^a=bc,d^a=d^52,e^a=e^58

Conjugacy classes & character table

6257 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₅₀₈
maximal normal subgroup of	Dic₁₂₅₀₈

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, 53, 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.4^52);;
gap> SetConjugate(coll, 5, 1, F.5^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6254 #72236043