Dic₁₃₉₄₀ #72290283

order	55760
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₉₄₀

Splittings

Hall	C₅×C₁₇×C₄₁ ⋊ Q₁₆
Hall	C₁₇×C₄₁ ⋊ Dic₂₀
Hall	C₅×C₄₁ ⋊ Dic₆₈
Hall	C₅×C₁₇ ⋊ Dic₁₆₄
Sylow	C₄₁ ⋊ Dic₃₄₀
Sylow	C₁₇ ⋊ Dic₈₂₀
Sylow	C₅ ⋊ Dic₂₇₈₈

Subgroups

	subgroup	G/H
center	C₂	D₁₃₉₄₀
commutator, cosocle	C₄×C₅×C₁₇×C₄₁	C₂²
Frattini	C₄	C₂×D₃₄₈₅
Fitting	C₅×C₈×C₁₇×C₄₁	C₂
socle, prime radical	C₂×C₅×C₁₇×C₄₁	D₄
hypercenter	C₈	D₃₄₈₅
hypocenter	C₅×C₁₇×C₄₁	Q₁₆
LCS	G ▷ C₄×C₅×C₁₇×C₄₁ ▷ C₂×C₅×C₁₇×C₄₁ ▷ C₅×C₁₇×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 55760

A minimal faithful permutation representation — 2 generators on 79 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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,21)(19,20)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2^4,5,17,41|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^16,g^a=g^40

Conjugacy classes & character table

13943 conjugacy classes. The character table is too large to display.

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, 5, 17, 41]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13940 #72290283