Dic₁₄₇₂₅ #72291687

order	58900
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₇₂₅

Splittings

derived, Hall	C₁₉×C₂₅×C₃₁ ⋊ C₄
Hall	C₂₅×C₃₁ ⋊ Dic₁₉
Hall	C₁₉×C₃₁ ⋊ Dic₂₅
Hall	C₁₉×C₂₅ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₄₇₅
Sylow	C₂₅ ⋊ Dic₅₈₉
Sylow	C₁₉ ⋊ Dic₇₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₄₇₂₅
commutator, hypocenter	C₁₉×C₂₅×C₃₁	C₄
Frattini	C₂×C₅	D₂₉₄₅
Fitting, cosocle	C₂×C₁₉×C₂₅×C₃₁	C₂
socle, prime radical	C₂×C₅×C₁₉×C₃₁	D₅
LCS	G ▷ C₁₉×C₂₅×C₃₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 58900

A minimal faithful permutation representation — 2 generators on 79 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);(1,3,2,4)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)

Polycyclic presentation:

2^2,5,19,31,5|a^2=b,c^5=f,c^a=c^4f^4,d^a=d^18,e^a=e^30,f^a=f^4

Conjugacy classes & character table

14728 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, 5, 19, 31, 5]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^4*F.6^4);;
gap> SetConjugate(coll, 4, 1, F.4^18);;
gap> SetConjugate(coll, 5, 1, F.5^30);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14725 #72291687