Dic₆₃₀₀ #72236070

order	25200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₃₀₀

Splittings

Hall	C₇×C₉×C₂₅ ⋊ Q₁₆
Hall	C₉×C₂₅ ⋊ Dic₂₈
Hall	C₇×C₂₅ ⋊ Dic₃₆
Hall	C₇×C₉ ⋊ Dic₁₀₀
Sylow	C₂₅ ⋊ Dic₂₅₂
Sylow	C₉ ⋊ Dic₇₀₀
Sylow	C₇ ⋊ Dic₉₀₀

Subgroups

	subgroup	G/H
center	C₂	D₆₃₀₀
commutator, cosocle	C₄×C₇×C₉×C₂₅	C₂²
Frattini	C₃×C₄×C₅	C₂×D₁₀₅
Fitting	C₇×C₈×C₉×C₂₅	C₂
socle, prime radical	C₂×C₃×C₅×C₇	D₆₀
hypercenter	C₈	D₁₅₇₅
hypocenter	C₇×C₉×C₂₅	Q₁₆
LCS	G ▷ C₄×C₇×C₉×C₂₅ ▷ C₂×C₇×C₉×C₂₅ ▷ C₇×C₉×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

162 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 25200

A minimal faithful permutation representation — 2 generators on 57 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,25)(19,24)(20,23)(21,22)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(52,57)(53,56)(54,55)

Polycyclic presentation:

2^4,3,5,7,3,5|a^2=d,b^2=cd,c^2=d,e^3=h,f^5=i,b^a=bc,c^a=cd,e^a=e^2h^2,f^a=f^4i^4,g^a=g^6,h^a=h^2,i^a=i^4

Conjugacy classes & character table

6303 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₆₀₀
maximal normal subgroup of	Dic₁₂₆₀₀

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 7, 3, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
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*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^4*F.9^4);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);