Dic₁₅₆₃₁ #72295367

order	62524
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₆₃₁

Splittings

derived, Hall	C₁₁×C₂₉×C₄₉ ⋊ C₄
Hall	C₂₉×C₄₉ ⋊ Dic₁₁
Hall	C₁₁×C₄₉ ⋊ Dic₂₉
Hall	C₁₁×C₂₉ ⋊ Dic₄₉
Sylow	C₄₉ ⋊ Dic₃₁₉
Sylow	C₂₉ ⋊ Dic₅₃₉
Sylow	C₁₁ ⋊ Dic₁₄₂₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₅₆₃₁
commutator, hypocenter	C₁₁×C₂₉×C₄₉	C₄
Frattini	C₂×C₇	D₂₂₃₃
Fitting, cosocle	C₂×C₁₁×C₂₉×C₄₉	C₂
socle, prime radical	C₂×C₇×C₁₁×C₂₉	D₇
LCS	G ▷ C₁₁×C₂₉×C₄₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 62524

A minimal faithful permutation representation — 2 generators on 93 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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3,2,4)(6,53)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(55,64)(56,63)(57,62)(58,61)(59,60)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80)

Polycyclic presentation:

2^2,7,11,29,7|a^2=b,c^7=f,c^a=c^6f^6,d^a=d^10,e^a=e^28,f^a=f^6

Conjugacy classes & character table

15634 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, 7, 11, 29, 7]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^6*F.6^6);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^28);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15631 #72295367