Dic₅₁₃ #72210307

order	2052
automorphism	Aut(G) order 332424
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₁₃

Splittings

derived, Hall	C₁₉×C₂₇ ⋊ C₄
Sylow	C₂₇ ⋊ Dic₁₉
Sylow	C₁₉ ⋊ Dic₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₁₃
commutator, hypocenter	C₁₉×C₂₇	C₄
Frattini	C₂×C₉	D₅₇
Fitting, cosocle	C₂×C₁₉×C₂₇	C₂
socle, prime radical	C₂×C₃×C₁₉	D₉
LCS	G ▷ C₁₉×C₂₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 2052

A minimal faithful permutation representation — 2 generators on 50 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);(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)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)

Polycyclic presentation:

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

Conjugacy classes & character table

516 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₁₀₂₆, Dic₁₅₃₉, Dic₂₅₆₅, …
maximal normal subgroup of	Dic₁₀₂₆
maximal quotient of	Dic₁₅₃₉, Dic₂₅₆₅, Dic₃₅₉₁, …

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, 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.4^18);;
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);

Dic513 #72210307