Dic₁₄₉₄₅ #72292297

order	59780
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₉₄₅

Splittings

derived, Hall	C₅×C₄₉×C₆₁ ⋊ C₄
Hall	C₄₉×C₆₁ ⋊ Dic₅
Hall	C₅×C₆₁ ⋊ Dic₄₉
Hall	C₅×C₄₉ ⋊ Dic₆₁
Sylow	C₆₁ ⋊ Dic₂₄₅
Sylow	C₄₉ ⋊ Dic₃₀₅
Sylow	C₅ ⋊ Dic₂₉₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₄₉₄₅
commutator, hypocenter	C₅×C₄₉×C₆₁	C₄
Frattini	C₂×C₇	D₂₁₃₅
Fitting, cosocle	C₂×C₅×C₄₉×C₆₁	C₂
socle, prime radical	C₂×C₅×C₇×C₆₁	D₇
LCS	G ▷ C₅×C₄₉×C₆₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 59780

A minimal faithful permutation representation — 2 generators on 119 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);(1,3,2,4)(6,9)(7,8)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(60,119)(61,118)(62,117)(63,116)(64,115)(65,114)(66,113)(67,112)(68,111)(69,110)(70,109)(71,108)(72,107)(73,106)(74,105)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)

Polycyclic presentation:

2^2,5,7,61,7|a^2=b,d^7=f,c^a=c^4,d^a=d^6f^6,e^a=e^60,f^a=f^6

Conjugacy classes & character table

14948 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, 5, 7, 61, 7]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^6*F.6^6);;
gap> SetConjugate(coll, 5, 1, F.5^60);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14945 #72292297