Dic₁₃₅₀₉ #72289585

order	54036
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₅₀₉

Splittings

derived, Hall	C₉×C₁₉×C₇₉ ⋊ C₄
Hall	C₁₉×C₇₉ ⋊ Dic₉
Hall	C₉×C₇₉ ⋊ Dic₁₉
Hall	C₉×C₁₉ ⋊ Dic₇₉
Sylow	C₇₉ ⋊ Dic₁₇₁
Sylow	C₁₉ ⋊ Dic₇₁₁
Sylow	C₉ ⋊ Dic₁₅₀₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₃₅₀₉
commutator, hypocenter	C₉×C₁₉×C₇₉	C₄
Frattini	C₂×C₃	D₄₅₀₃
Fitting, cosocle	C₂×C₉×C₁₉×C₇₉	C₂
socle, prime radical	C₂×C₃×C₁₉×C₇₉	D₃
LCS	G ▷ C₉×C₁₉×C₇₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 54036

A minimal faithful permutation representation — 2 generators on 111 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);(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)(34,111)(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:

2^2,3,19,79,3|a^2=b,c^3=f,c^a=c^2f^2,d^a=d^18,e^a=e^78,f^a=f^2

Conjugacy classes & character table

13512 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, 79, 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.5^78);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13509 #72289585