Dic₁₂₅₀₂ #72286740

order	50008
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₅₀₂

Splittings

Hall	C₇×C₁₉×C₄₇ ⋊ Q₈
Hall	C₁₉×C₄₇ ⋊ Dic₁₄
Hall	C₇×C₄₇ ⋊ Dic₃₈
Hall	C₇×C₁₉ ⋊ Dic₉₄
Sylow	C₄₇ ⋊ Dic₂₆₆
Sylow	C₁₉ ⋊ Dic₆₅₈
Sylow	C₇ ⋊ Dic₁₇₈₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₂₅₁
commutator, socle, prime radical, cosocle	C₂×C₇×C₁₉×C₄₇	C₂²
Fitting	C₄×C₇×C₁₉×C₄₇	C₂
hypercenter	C₄	D₆₂₅₁
hypocenter	C₇×C₁₉×C₄₇	Q₈
LCS	G ▷ C₂×C₇×C₁₉×C₄₇ ▷ C₇×C₁₉×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 50008

A minimal faithful permutation representation — 2 generators on 81 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)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(1,5,3,7)(2,6,4,8)(10,15)(11,14)(12,13)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(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:

2^3,7,19,47|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^18,f^a=f^46

Conjugacy classes & character table

12505 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, 7, 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.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12502 #72286740