Dic₄₄₁₀ #72215877

order	17640
automorphism	Aut(G) order 17781120
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₄₁₀

Splittings

Hall	C₅×C₉×C₄₉ ⋊ Q₈
Hall	C₉×C₄₉ ⋊ Dic₁₀
Hall	C₅×C₄₉ ⋊ Dic₁₈
Sylow	C₄₉ ⋊ Dic₉₀
Hall	C₅×C₉ ⋊ Dic₉₈
Sylow	C₉ ⋊ Dic₄₉₀
Sylow	C₅ ⋊ Dic₈₈₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₂₀₅
commutator, cosocle	C₂×C₅×C₉×C₄₉	C₂²
Frattini	C₂×C₃×C₇	C₂×D₁₀₅
Fitting	C₄×C₅×C₉×C₄₉	C₂
socle, prime radical	C₂×C₃×C₅×C₇	C₂×D₂₁
hypercenter	C₄	D₂₂₀₅
hypocenter	C₅×C₉×C₄₉	Q₈
LCS	G ▷ C₂×C₅×C₉×C₄₉ ▷ C₅×C₉×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 17640

A minimal faithful permutation representation — 2 generators on 71 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);(1,5,3,7)(2,6,4,8)(10,17)(11,16)(12,15)(13,14)(19,22)(20,21)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

4413 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₈₂₀, Dic₁₃₂₃₀
maximal normal subgroup of	Dic₈₈₂₀
maximal quotient of	Dic₁₃₂₃₀

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 7, 3, 7]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^6*F.8^6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4410 #72215877