Dic₃₆₉₀ #72214834

order	14760
automorphism	Aut(G) order 14169600
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₆₉₀

Splittings

Hall	C₅×C₉×C₄₁ ⋊ Q₈
Hall	C₉×C₄₁ ⋊ Dic₁₀
Hall	C₅×C₄₁ ⋊ Dic₁₈
Hall	C₅×C₉ ⋊ Dic₈₂
Sylow	C₄₁ ⋊ Dic₉₀
Sylow	C₉ ⋊ Dic₄₁₀
Sylow	C₅ ⋊ Dic₇₃₈

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₈₄₅
commutator, cosocle	C₂×C₅×C₉×C₄₁	C₂²
Frattini	C₂×C₃	C₂×D₆₁₅
Fitting	C₄×C₅×C₉×C₄₁	C₂
socle, prime radical	C₂×C₃×C₅×C₄₁	C₂×D₃
hypercenter	C₄	D₁₈₄₅
hypocenter	C₅×C₉×C₄₁	Q₈
LCS	G ▷ C₂×C₅×C₉×C₄₁ ▷ C₅×C₉×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 14760

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

Polycyclic presentation:

2^3,3,5,41,3|a^2=c,b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^4,f^a=f^40,g^a=g^2

Conjugacy classes & character table

3693 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₃₈₀, Dic₁₁₀₇₀
maximal normal subgroup of	Dic₇₃₈₀
maximal quotient of	Dic₁₁₀₇₀

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, 3, 5, 41, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
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.6^40);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3690 #72214834