Dic₁₄₃₁ #72211951

order	5724
automorphism	Aut(G) order 2678832
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₃₁

Splittings

derived, Hall	C₂₇×C₅₃ ⋊ C₄
Sylow	C₅₃ ⋊ Dic₂₇
Sylow	C₂₇ ⋊ Dic₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₄₃₁
commutator, hypocenter	C₂₇×C₅₃	C₄
Frattini	C₂×C₉	D₁₅₉
Fitting, cosocle	C₂×C₂₇×C₅₃	C₂
socle, prime radical	C₂×C₃×C₅₃	D₉
LCS	G ▷ C₂₇×C₅₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 5724

A minimal faithful permutation representation — 2 generators on 84 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);(1,3,2,4)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Polycyclic presentation:

2^2,3,53,3^2|a^2=b,c^3=e,e^3=f,c^a=c^2e^2f^2,d^a=d^52,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

1434 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₂₈₆₂, Dic₄₂₉₃, Dic₇₁₅₅, …
maximal normal subgroup of	Dic₂₈₆₂
maximal quotient of	Dic₄₂₉₃, Dic₇₁₅₅, Dic₁₀₀₁₇, …

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, 53, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 4, 1, F.4^52);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1431 #72211951