Dic₄₃₈₀ #72215814

order	17520
automorphism	Aut(G) order 20183040
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₃₈₀

Splittings

Hall	C₃×C₅×C₇₃ ⋊ Q₁₆
Hall	C₅×C₇₃ ⋊ Dic₁₂
Hall	C₃×C₇₃ ⋊ Dic₂₀
Sylow	C₇₃ ⋊ Dic₆₀
Hall	C₃×C₅ ⋊ Dic₂₉₂
Sylow	C₅ ⋊ Dic₈₇₆
Sylow	C₃ ⋊ Dic₁₄₆₀

Subgroups

	subgroup	G/H
center	C₂	D₄₃₈₀
commutator, cosocle	C₃×C₄×C₅×C₇₃	C₂²
Frattini	C₄	C₂×D₁₀₉₅
Fitting	C₃×C₅×C₈×C₇₃	C₂
socle, prime radical	C₂×C₃×C₅×C₇₃	D₄
hypercenter	C₈	D₁₀₉₅
hypocenter	C₃×C₅×C₇₃	Q₁₆
LCS	G ▷ C₃×C₄×C₅×C₇₃ ▷ C₂×C₃×C₅×C₇₃ ▷ C₃×C₅×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 17520

A minimal faithful permutation representation — 2 generators on 97 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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,24)(22,23)(26,97)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)

Polycyclic presentation:

2^4,3,5,73|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^72

Conjugacy classes & character table

4383 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₇₆₀, Dic₁₃₁₄₀
maximal normal subgroup of	Dic₈₇₆₀
maximal quotient of	Dic₁₃₁₄₀

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, 3, 5, 73]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4380 #72215814