D₂₈₆₅₂ #72292269

order	57304
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₈₆₅₂

Splittings

Fitting	C₄×C₁₃×C₁₉×C₂₉ ⋊ C₂
Hall	C₁₃×C₁₉×C₂₉ ⋊ D₄
Hall	C₁₉×C₂₉ ⋊ D₅₂
Hall	C₁₃×C₂₉ ⋊ D₇₆
Hall	C₁₃×C₁₉ ⋊ D₁₁₆
Sylow	C₂₉ ⋊ D₉₈₈
Sylow	C₁₉ ⋊ D₁₅₀₈
Sylow	C₁₃ ⋊ D₂₂₀₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₁₆₃
commutator, socle, cosocle	C₂×C₁₃×C₁₉×C₂₉	C₂²
Fitting	C₄×C₁₃×C₁₉×C₂₉	C₂
hypercenter	C₄	D₇₁₆₃
hypocenter	C₁₃×C₁₉×C₂₉	D₄
LCS	G ▷ C₂×C₁₃×C₁₉×C₂₉ ▷ C₁₃×C₁₉×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 28652

A minimal faithful permutation representation — 2 generators on 65 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);(2,4)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2^3,13,19,29|b^2=c,b^a=bc,d^a=d^12,e^a=e^18,f^a=f^28

Conjugacy classes & character table

14329 conjugacy classes. The character table is too large to display.

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, 13, 19, 29]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D28652 #72292269