Dic₇₄₈₀ #72236617

order	29920
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₄₈₀

Splittings

Hall	C₅×C₁₁×C₁₇ ⋊ Q₃₂
Hall	C₁₁×C₁₇ ⋊ Dic₄₀
Hall	C₅×C₁₇ ⋊ Dic₈₈
Hall	C₅×C₁₁ ⋊ Dic₁₃₆
Sylow	C₁₇ ⋊ Dic₄₄₀
Sylow	C₁₁ ⋊ Dic₆₈₀
Sylow	C₅ ⋊ Dic₁₄₉₆

Subgroups

	subgroup	G/H
center	C₂	D₇₄₈₀
commutator, cosocle	C₅×C₈×C₁₁×C₁₇	C₂²
Frattini	C₈	C₂×D₉₃₅
Fitting	C₅×C₁₁×C₁₆×C₁₇	C₂
socle, prime radical	C₂×C₅×C₁₁×C₁₇	D₈
hypercenter	C₁₆	D₉₃₅
hypocenter	C₅×C₁₁×C₁₇	Q₃₂
LCS	G ▷ C₅×C₈×C₁₁×C₁₇ ▷ C₄×C₅×C₁₁×C₁₇ ▷ C₂×C₅×C₁₁×C₁₇ ▷ C₅×C₁₁×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 29920

A minimal faithful permutation representation — 2 generators on 65 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,37)(35,36)(39,48)(40,47)(41,46)(42,45)(43,44)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)

Polycyclic presentation:

2^5,5,11,17|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^4,g^a=g^10,h^a=h^16

Conjugacy classes & character table

7483 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₉₆₀
maximal normal subgroup of	Dic₁₄₉₆₀

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, 2, 5, 11, 17]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> SetConjugate(coll, 8, 1, F.8^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7480 #72236617