D₂₆₅₂ #72210532

order	5304
automorphism	Aut(G) order 2036736
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₅₂

Splittings

Fitting	C₃×C₄×C₁₃×C₁₇ ⋊ C₂
Hall	C₃×C₁₃×C₁₇ ⋊ D₄
Hall	C₁₃×C₁₇ ⋊ D₁₂
Hall	C₃×C₁₇ ⋊ D₅₂
Hall	C₃×C₁₃ ⋊ D₆₈
Sylow	C₁₇ ⋊ D₁₅₆
Sylow	C₁₃ ⋊ D₂₀₄
Sylow	C₃ ⋊ D₈₈₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₆₃
commutator, socle, cosocle	C₂×C₃×C₁₃×C₁₇	C₂²
Fitting	C₃×C₄×C₁₃×C₁₇	C₂
hypercenter	C₄	D₆₆₃
hypocenter	C₃×C₁₃×C₁₇	D₄
LCS	G ▷ C₂×C₃×C₁₃×C₁₇ ▷ C₃×C₁₃×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 2652

A minimal faithful permutation representation — 2 generators on 37 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);(2,4)(6,7)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)

Polycyclic presentation:

2^3,3,13,17|b^2=c,b^a=bc,d^a=d^2,e^a=e^12,f^a=f^16

Conjugacy classes & character table

1329 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₃₀₄, D₇₉₅₆, D₁₃₂₆₀, …
maximal normal subgroup of	D₅₃₀₄
maximal quotient of	D₅₃₀₄, Dic₂₆₅₂, D₇₉₅₆, …

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 13, 17]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D2652 #72210532