D₁₅₁₂₀ #72271815

order	30240
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₅₁₂₀

Splittings

Fitting	C₅×C₇×C₁₆×C₂₇ ⋊ C₂
Hall	C₅×C₇×C₂₇ ⋊ D₁₆
Hall	C₇×C₂₇ ⋊ D₈₀
Hall	C₅×C₂₇ ⋊ D₁₁₂
Hall	C₅×C₇ ⋊ D₄₃₂
Sylow	C₂₇ ⋊ D₅₆₀
Sylow	C₇ ⋊ D₂₁₆₀
Sylow	C₅ ⋊ D₃₀₂₄

Subgroups

	subgroup	G/H
center	C₂	D₇₅₆₀
commutator, cosocle	C₅×C₇×C₈×C₂₇	C₂²
Frattini	C₈×C₉	C₂×D₁₀₅
Fitting	C₅×C₇×C₁₆×C₂₇	C₂
socle	C₂×C₃×C₅×C₇	D₇₂
hypercenter	C₁₆	D₉₄₅
hypocenter	C₅×C₇×C₂₇	D₁₆
LCS	G ▷ C₅×C₇×C₈×C₂₇ ▷ C₄×C₅×C₇×C₂₇ ▷ C₂×C₅×C₇×C₂₇ ▷ C₅×C₇×C₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

224 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 83 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) = 55
transitive degree	μ_t(G) = 15120

A minimal faithful permutation representation — 2 generators on 55 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(45,48)(46,47)(50,55)(51,54)(52,53)

Polycyclic presentation:

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

Conjugacy classes & character table

7563 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₀₂₄₀
maximal normal subgroup of	D₃₀₂₄₀
maximal quotient of	Dic₁₅₁₂₀, D₃₀₂₄₀

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 5, 7, 3, 3]);;
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.9);;
gap> SetPower(coll, 9, F.10);;
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.9^2*F.10^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 9, 1, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);