D₁₅₂₈₈ #72272007

order	30576
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₅₂₈₈

Splittings

Fitting	C₃×C₈×C₁₃×C₄₉ ⋊ C₂
Hall	C₃×C₁₃×C₄₉ ⋊ D₈
Hall	C₁₃×C₄₉ ⋊ D₂₄
Hall	C₃×C₄₉ ⋊ D₁₀₄
Sylow	C₄₉ ⋊ D₃₁₂
Hall	C₃×C₁₃ ⋊ D₃₉₂
Sylow	C₁₃ ⋊ D₁₁₇₆
Sylow	C₃ ⋊ D₅₀₉₆

Subgroups

	subgroup	G/H
center	C₂	D₇₆₄₄
commutator, cosocle	C₃×C₄×C₁₃×C₄₉	C₂²
Frattini	C₄×C₇	C₂×D₂₇₃
Fitting	C₃×C₈×C₁₃×C₄₉	C₂
socle	C₂×C₃×C₇×C₁₃	D₂₈
hypercenter	C₈	D₁₉₁₁
hypocenter	C₃×C₁₃×C₄₉	D₈
LCS	G ▷ C₃×C₄×C₁₃×C₄₉ ▷ C₂×C₃×C₁₃×C₄₉ ▷ C₃×C₁₃×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 15288

A minimal faithful permutation representation — 2 generators on 73 points:

(1,2,3,4,5,6,7,8)(9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73);(2,8)(3,7)(4,6)(10,11)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Polycyclic presentation:

2^4,3,7,13,7|b^2=cd,c^2=d,f^7=h,b^a=bc,c^a=cd,e^a=e^2,f^a=f^6h^6,g^a=g^12,h^a=h^6

Conjugacy classes & character table

7647 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₀₅₇₆
maximal normal subgroup of	D₃₀₅₇₆
maximal quotient of	Dic₁₅₂₈₈, D₃₀₅₇₆

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, 3, 7, 13, 7]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
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, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^6*F.8^6);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D15288 #72272007