Dic₁₃₆₁₇ #72289768

order	54468
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₆₁₇

Splittings

derived, Hall	C₉×C₁₇×C₈₉ ⋊ C₄
Hall	C₁₇×C₈₉ ⋊ Dic₉
Hall	C₉×C₈₉ ⋊ Dic₁₇
Hall	C₉×C₁₇ ⋊ Dic₈₉
Sylow	C₈₉ ⋊ Dic₁₅₃
Sylow	C₁₇ ⋊ Dic₈₀₁
Sylow	C₉ ⋊ Dic₁₅₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₃₆₁₇
commutator, hypocenter	C₉×C₁₇×C₈₉	C₄
Frattini	C₂×C₃	D₄₅₃₉
Fitting, cosocle	C₂×C₉×C₁₇×C₈₉	C₂
socle, prime radical	C₂×C₃×C₁₇×C₈₉	D₃
LCS	G ▷ C₉×C₁₇×C₈₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 54468

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

Polycyclic presentation:

2^2,3,17,89,3|a^2=b,c^3=f,c^a=c^2f^2,d^a=d^16,e^a=e^88,f^a=f^2

Conjugacy classes & character table

13620 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, 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.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^88);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13617 #72289768