Dic₆₀₀ #72210442

order	2400
automorphism	Aut(G) order 384000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₀₀

Splittings

Hall	C₃×C₂₅ ⋊ Q₃₂
Sylow	C₂₅ ⋊ Dic₂₄
Sylow	C₃ ⋊ Dic₂₀₀

Subgroups

	subgroup	G/H
center	C₂	D₆₀₀
commutator, cosocle	C₃×C₈×C₂₅	C₂²
Frattini	C₅×C₈	C₂×D₁₅
Fitting	C₃×C₁₆×C₂₅	C₂
socle, prime radical	C₂×C₃×C₅	D₄₀
hypercenter	C₁₆	D₇₅
hypocenter	C₃×C₂₅	Q₃₂
LCS	G ▷ C₃×C₈×C₂₅ ▷ C₃×C₄×C₂₅ ▷ C₂×C₃×C₂₅ ▷ C₃×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 2400

A minimal faithful permutation representation — 2 generators on 60 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);(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)(34,35)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

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

Conjugacy classes & character table

603 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₁₂₀₀, Dic₁₈₀₀, Dic₃₀₀₀, …
maximal normal subgroup of	Dic₁₂₀₀
maximal quotient of	Dic₁₈₀₀, Dic₃₀₀₀, Dic₄₂₀₀, …

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, 5, 5]);;
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, 7, 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);;
gap> SetConjugate(coll, 7, 1, F.7^4*F.8^4);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic600 #72210442