Dic₅₀₄ #72210309

order	2016
automorphism	Aut(G) order 290304
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₀₄

Splittings

Hall	C₇×C₉ ⋊ Q₃₂
Sylow	C₉ ⋊ Dic₅₆
Sylow	C₇ ⋊ Dic₇₂

Subgroups

	subgroup	G/H
center	C₂	D₅₀₄
commutator, cosocle	C₇×C₈×C₉	C₂²
Frattini	C₃×C₈	C₂×D₂₁
Fitting	C₇×C₉×C₁₆	C₂
socle, prime radical	C₂×C₃×C₇	D₂₄
hypercenter	C₁₆	D₆₃
hypocenter	C₇×C₉	Q₃₂
LCS	G ▷ C₇×C₈×C₉ ▷ C₄×C₇×C₉ ▷ C₂×C₇×C₉ ▷ C₇×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 2016

A minimal faithful permutation representation — 2 generators on 48 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);(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)(34,41)(35,40)(36,39)(37,38)(43,48)(44,47)(45,46)

Polycyclic presentation:

2^5,3,7,3|a^2=e,b^2=cd,c^2=de,d^2=e,f^3=h,b^a=bc,c^a=cd,d^a=de,f^a=f^2h^2,g^a=g^6,h^a=h^2

Conjugacy classes & character table

507 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₁₀₀₈, Dic₁₅₁₂, Dic₂₅₂₀, …
maximal normal subgroup of	Dic₁₀₀₈
maximal quotient of	Dic₁₅₁₂, Dic₂₅₂₀, Dic₃₅₂₈, …

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

Dic504 #72210309