Dic₁₄₃₁₉ #72290924

order	57276
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₃₁₉

Splittings

derived, Hall	C₉×C₃₇×C₄₃ ⋊ C₄
Hall	C₃₇×C₄₃ ⋊ Dic₉
Hall	C₉×C₄₃ ⋊ Dic₃₇
Hall	C₉×C₃₇ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₃₃₃
Sylow	C₃₇ ⋊ Dic₃₈₇
Sylow	C₉ ⋊ Dic₁₅₉₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₄₃₁₉
commutator, hypocenter	C₉×C₃₇×C₄₃	C₄
Frattini	C₂×C₃	D₄₇₇₃
Fitting, cosocle	C₂×C₉×C₃₇×C₄₃	C₂
socle, prime radical	C₂×C₃×C₃₇×C₄₃	D₃
LCS	G ▷ C₉×C₃₇×C₄₃
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 57276

A minimal faithful permutation representation — 2 generators on 93 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(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:

2^2,3,37,43,3|a^2=b,c^3=f,c^a=c^2f^2,d^a=d^36,e^a=e^42,f^a=f^2

Conjugacy classes & character table

14322 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, 37, 43, 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.4^36);;
gap> SetConjugate(coll, 5, 1, F.5^42);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14319 #72290924