Dic₂₆₈₀ #72212901

order	10720
automorphism	Aut(G) order 11320320
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₆₈₀

Splittings

Hall	C₅×C₆₇ ⋊ Q₃₂
Sylow	C₆₇ ⋊ Dic₄₀
Sylow	C₅ ⋊ Dic₅₃₆

Subgroups

	subgroup	G/H
center	C₂	D₂₆₈₀
commutator, cosocle	C₅×C₈×C₆₇	C₂²
Frattini	C₈	C₂×D₃₃₅
Fitting	C₅×C₁₆×C₆₇	C₂
socle, prime radical	C₂×C₅×C₆₇	D₈
hypercenter	C₁₆	D₃₃₅
hypocenter	C₅×C₆₇	Q₃₂
LCS	G ▷ C₅×C₈×C₆₇ ▷ C₄×C₅×C₆₇ ▷ C₂×C₅×C₆₇ ▷ C₅×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 104
transitive degree	μ_t(G) = 10720

A minimal faithful permutation representation — 2 generators on 104 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)(38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104);(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)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2^5,5,67|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^66

Conjugacy classes & character table

2683 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₃₆₀, Dic₈₀₄₀, Dic₁₃₄₀₀
maximal normal subgroup of	Dic₅₃₆₀
maximal quotient of	Dic₈₀₄₀, Dic₁₃₄₀₀

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, 2, 2, 5, 67]);;
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.7^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2680 #72212901