Dic₁₅₂₁₉ #72292779

order	60876
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₂₁₉

Splittings

derived, Hall	C₉×C₁₉×C₈₉ ⋊ C₄
Hall	C₁₉×C₈₉ ⋊ Dic₉
Hall	C₉×C₈₉ ⋊ Dic₁₉
Hall	C₉×C₁₉ ⋊ Dic₈₉
Sylow	C₈₉ ⋊ Dic₁₇₁
Sylow	C₁₉ ⋊ Dic₈₀₁
Sylow	C₉ ⋊ Dic₁₆₉₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₅₂₁₉
commutator, hypocenter	C₉×C₁₉×C₈₉	C₄
Frattini	C₂×C₃	D₅₀₇₃
Fitting, cosocle	C₂×C₉×C₁₉×C₈₉	C₂
socle, prime radical	C₂×C₃×C₁₉×C₈₉	D₃
LCS	G ▷ C₉×C₁₉×C₈₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 60876

A minimal faithful permutation representation — 2 generators on 121 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(34,121)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)

Polycyclic presentation:

2^2,3,19,89,3|a^2=b,c^3=f,c^a=c^2f^2,d^a=d^18,e^a=e^88,f^a=f^2

Conjugacy classes & character table

15222 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, 3, 19, 89, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.6^2);;
gap> SetConjugate(coll, 4, 1, F.4^18);;
gap> SetConjugate(coll, 5, 1, F.5^88);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15219 #72292779