Dic₁₄₂₁₀ #72290719

order	56840
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₂₁₀

Splittings

Hall	C₅×C₂₉×C₄₉ ⋊ Q₈
Hall	C₂₉×C₄₉ ⋊ Dic₁₀
Hall	C₅×C₄₉ ⋊ Dic₅₈
Hall	C₅×C₂₉ ⋊ Dic₉₈
Sylow	C₄₉ ⋊ Dic₂₉₀
Sylow	C₂₉ ⋊ Dic₄₉₀
Sylow	C₅ ⋊ Dic₂₈₄₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₁₀₅
commutator, cosocle	C₂×C₅×C₂₉×C₄₉	C₂²
Frattini	C₂×C₇	C₂×D₁₀₁₅
Fitting	C₄×C₅×C₂₉×C₄₉	C₂
socle, prime radical	C₂×C₅×C₇×C₂₉	C₂×D₇
hypercenter	C₄	D₇₁₀₅
hypocenter	C₅×C₂₉×C₄₉	Q₈
LCS	G ▷ C₂×C₅×C₂₉×C₄₉ ▷ C₅×C₂₉×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 56840

A minimal faithful permutation representation — 2 generators on 91 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)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)

Polycyclic presentation:

2^3,5,7,29,7|a^2=c,b^2=c,e^7=g,b^a=bc,d^a=d^4,e^a=e^6g^6,f^a=f^28,g^a=g^6

Conjugacy classes & character table

14213 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, 2, 5, 7, 29, 7]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^6*F.7^6);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14210 #72290719