Dic₁₅₉₀₃ #72296154

order	63612
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₉₀₃

Splittings

derived, Hall	C₁₉×C₂₇×C₃₁ ⋊ C₄
Hall	C₂₇×C₃₁ ⋊ Dic₁₉
Hall	C₁₉×C₃₁ ⋊ Dic₂₇
Hall	C₁₉×C₂₇ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₅₁₃
Sylow	C₂₇ ⋊ Dic₅₈₉
Sylow	C₁₉ ⋊ Dic₈₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₅₉₀₃
commutator, hypocenter	C₁₉×C₂₇×C₃₁	C₄
Frattini	C₂×C₉	D₁₇₆₇
Fitting, cosocle	C₂×C₁₉×C₂₇×C₃₁	C₂
socle, prime radical	C₂×C₃×C₁₉×C₃₁	D₉
LCS	G ▷ C₁₉×C₂₇×C₃₁
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 63612

A minimal faithful permutation representation — 2 generators on 81 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);(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)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Polycyclic presentation:

2^2,3,19,31,3^2|a^2=b,c^3=f,f^3=g,c^a=c^2f^2g^2,d^a=d^18,e^a=e^30,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

15906 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, 3, 19, 31, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 4, 1, F.4^18);;
gap> SetConjugate(coll, 5, 1, F.5^30);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15903 #72296154