D₂₈₄₁₃ #72291941

order	56826
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₈₄₁₃

Splittings

Fitting, derived, Hall	C₇×C₉×C₁₁×C₄₁ ⋊ C₂
Hall	C₉×C₁₁×C₄₁ ⋊ D₇
Hall	C₇×C₁₁×C₄₁ ⋊ D₉
Hall	C₇×C₉×C₄₁ ⋊ D₁₁
Hall	C₇×C₉×C₁₁ ⋊ D₄₁
Hall	C₁₁×C₄₁ ⋊ D₆₃
Hall	C₉×C₄₁ ⋊ D₇₇
Hall	C₇×C₄₁ ⋊ D₉₉
Hall	C₉×C₁₁ ⋊ D₂₈₇
Hall	C₇×C₁₁ ⋊ D₃₆₉
Hall	C₇×C₉ ⋊ D₄₅₁
Sylow	C₄₁ ⋊ D₆₉₃
Sylow	C₁₁ ⋊ D₂₅₈₃
Sylow	C₉ ⋊ D₃₁₅₇
Sylow	C₇ ⋊ D₄₀₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₉×C₁₁×C₄₁	C₂
Frattini	C₃	D₉₄₇₁
socle	C₃×C₇×C₁₁×C₄₁	D₃
LCS	G ▷ C₇×C₉×C₁₁×C₄₁
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 28413

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

Polycyclic presentation:

2,3,7,11,41,3|b^3=f,b^a=b^2f^2,c^a=c^6,d^a=d^10,e^a=e^40,f^a=f^2

Conjugacy classes & character table

14208 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, 3, 7, 11, 41, 3]);;
gap> SetPower(coll, 2, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^40);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D28413 #72291941