Dic₈₉₃₀ #72273223

order	35720
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₈₉₃₀

Splittings

Hall	C₅×C₁₉×C₄₇ ⋊ Q₈
Hall	C₁₉×C₄₇ ⋊ Dic₁₀
Hall	C₅×C₄₇ ⋊ Dic₃₈
Hall	C₅×C₁₉ ⋊ Dic₉₄
Sylow	C₄₇ ⋊ Dic₁₉₀
Sylow	C₁₉ ⋊ Dic₄₇₀
Sylow	C₅ ⋊ Dic₁₇₈₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₄₄₆₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₁₉×C₄₇	C₂²
Fitting	C₄×C₅×C₁₉×C₄₇	C₂
hypercenter	C₄	D₄₄₆₅
hypocenter	C₅×C₁₉×C₄₇	Q₈
LCS	G ▷ C₂×C₅×C₁₉×C₄₇ ▷ C₅×C₁₉×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 35720

A minimal faithful permutation representation — 2 generators on 79 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2^3,5,19,47|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^18,f^a=f^46

Conjugacy classes & character table

8933 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, 2, 5, 19, 47]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic8930 #72273223