D₃₂₆₈₃ #72304196

order	65366
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₂₆₈₃

Splittings

Fitting, derived, Hall	C₂₃×C₂₉×C₄₉ ⋊ C₂
Hall	C₂₉×C₄₉ ⋊ D₂₃
Hall	C₂₃×C₄₉ ⋊ D₂₉
Hall	C₂₃×C₂₉ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₆₆₇
Sylow	C₂₉ ⋊ D₁₁₂₇
Sylow	C₂₃ ⋊ D₁₄₂₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₃×C₂₉×C₄₉	C₂
Frattini	C₇	D₄₆₆₉
socle	C₇×C₂₃×C₂₉	D₇
LCS	G ▷ C₂₃×C₂₉×C₄₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 32683

A minimal faithful permutation representation — 2 generators on 101 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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)

Polycyclic presentation:

2,7,23,29,7|b^7=e,b^a=b^6e^6,c^a=c^22,d^a=d^28,e^a=e^6

Conjugacy classes & character table

16343 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");;
gap> coll := SingleCollector(F, [2, 7, 23, 29, 7]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^6*F.5^6);;
gap> SetConjugate(coll, 3, 1, F.3^22);;
gap> SetConjugate(coll, 4, 1, F.4^28);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32683 #72304196