Dic₁₂₆₉₉ #72288075

order	50796
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₆₉₉

Splittings

derived, Hall	C₉×C₁₇×C₈₃ ⋊ C₄
Hall	C₁₇×C₈₃ ⋊ Dic₉
Hall	C₉×C₈₃ ⋊ Dic₁₇
Hall	C₉×C₁₇ ⋊ Dic₈₃
Sylow	C₈₃ ⋊ Dic₁₅₃
Sylow	C₁₇ ⋊ Dic₇₄₇
Sylow	C₉ ⋊ Dic₁₄₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₂₆₉₉
commutator, hypocenter	C₉×C₁₇×C₈₃	C₄
Frattini	C₂×C₃	D₄₂₃₃
Fitting, cosocle	C₂×C₉×C₁₇×C₈₃	C₂
socle, prime radical	C₂×C₃×C₁₇×C₈₃	D₃
LCS	G ▷ C₉×C₁₇×C₈₃
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 50796

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

Polycyclic presentation:

2^2,3,17,83,3|a^2=b,c^3=f,c^a=c^2f^2,d^a=d^16,e^a=e^82,f^a=f^2

Conjugacy classes & character table

12702 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, 17, 83, 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.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^82);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12699 #72288075