Dic₇₃₈₀ #72236564

order	29520
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₃₈₀

Splittings

Hall	C₅×C₉×C₄₁ ⋊ Q₁₆
Hall	C₉×C₄₁ ⋊ Dic₂₀
Hall	C₅×C₄₁ ⋊ Dic₃₆
Hall	C₅×C₉ ⋊ Dic₁₆₄
Sylow	C₄₁ ⋊ Dic₁₈₀
Sylow	C₉ ⋊ Dic₈₂₀
Sylow	C₅ ⋊ Dic₁₄₇₆

Subgroups

	subgroup	G/H
center	C₂	D₇₃₈₀
commutator, cosocle	C₄×C₅×C₉×C₄₁	C₂²
Frattini	C₃×C₄	C₂×D₆₁₅
Fitting	C₅×C₈×C₉×C₄₁	C₂
socle, prime radical	C₂×C₃×C₅×C₄₁	D₁₂
hypercenter	C₈	D₁₈₄₅
hypocenter	C₅×C₉×C₄₁	Q₁₆
LCS	G ▷ C₄×C₅×C₉×C₄₁ ▷ C₂×C₅×C₉×C₄₁ ▷ C₅×C₉×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 29520

A minimal faithful permutation representation — 2 generators on 71 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,25)(19,24)(20,23)(21,22)(27,30)(28,29)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

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

Conjugacy classes & character table

7383 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₇₆₀
maximal normal subgroup of	Dic₁₄₇₆₀

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, 2, 3, 5, 41, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^40);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7380 #72236564