D₂₆₁₁₇ #72289331

order	52234
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₆₁₁₇

Splittings

Fitting, derived, Hall	C₁₃×C₄₁×C₄₉ ⋊ C₂
Hall	C₄₁×C₄₉ ⋊ D₁₃
Hall	C₁₃×C₄₉ ⋊ D₄₁
Hall	C₁₃×C₄₁ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₅₃₃
Sylow	C₄₁ ⋊ D₆₃₇
Sylow	C₁₃ ⋊ D₂₀₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₃×C₄₁×C₄₉	C₂
Frattini	C₇	D₃₇₃₁
socle	C₇×C₁₃×C₄₁	D₇
LCS	G ▷ C₁₃×C₄₁×C₄₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 26117

A minimal faithful permutation representation — 2 generators on 103 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)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(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)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)

Polycyclic presentation:

2,7,13,41,7|b^7=e,b^a=b^6e^6,c^a=c^12,d^a=d^40,e^a=e^6

Conjugacy classes & character table

13060 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, 13, 41, 7]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^6*F.5^6);;
gap> SetConjugate(coll, 3, 1, F.3^12);;
gap> SetConjugate(coll, 4, 1, F.4^40);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26117 #72289331