Dic₁₃₇₆₄ #72289996

order	55056
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₇₆₄

Splittings

Hall	C₃×C₃₁×C₃₇ ⋊ Q₁₆
Hall	C₃₁×C₃₇ ⋊ Dic₁₂
Hall	C₃×C₃₇ ⋊ Dic₁₂₄
Hall	C₃×C₃₁ ⋊ Dic₁₄₈
Sylow	C₃₇ ⋊ Dic₃₇₂
Sylow	C₃₁ ⋊ Dic₄₄₄
Sylow	C₃ ⋊ Dic₄₅₈₈

Subgroups

	subgroup	G/H
center	C₂	D₁₃₇₆₄
commutator, cosocle	C₃×C₄×C₃₁×C₃₇	C₂²
Frattini	C₄	C₂×D₃₄₄₁
Fitting	C₃×C₈×C₃₁×C₃₇	C₂
socle, prime radical	C₂×C₃×C₃₁×C₃₇	D₄
hypercenter	C₈	D₃₄₄₁
hypocenter	C₃×C₃₁×C₃₇	Q₁₆
LCS	G ▷ C₃×C₄×C₃₁×C₃₇ ▷ C₂×C₃×C₃₁×C₃₇ ▷ C₃×C₃₁×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 55056

A minimal faithful permutation representation — 2 generators on 87 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Polycyclic presentation:

2^4,3,31,37|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^30,g^a=g^36

Conjugacy classes & character table

13767 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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 31, 37]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^30);;
gap> SetConjugate(coll, 7, 1, F.7^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13764 #72289996