D₂₅₂₅₆ #72288435

order	50512
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₅₂₅₆

Splittings

Fitting	C₇×C₈×C₁₁×C₄₁ ⋊ C₂
Hall	C₇×C₁₁×C₄₁ ⋊ D₈
Hall	C₁₁×C₄₁ ⋊ D₅₆
Hall	C₇×C₄₁ ⋊ D₈₈
Hall	C₇×C₁₁ ⋊ D₃₂₈
Sylow	C₄₁ ⋊ D₆₁₆
Sylow	C₁₁ ⋊ D₂₂₉₆
Sylow	C₇ ⋊ D₃₆₀₈

Subgroups

	subgroup	G/H
center	C₂	D₁₂₆₂₈
commutator, cosocle	C₄×C₇×C₁₁×C₄₁	C₂²
Frattini	C₄	C₂×D₃₁₅₇
Fitting	C₇×C₈×C₁₁×C₄₁	C₂
socle	C₂×C₇×C₁₁×C₄₁	D₄
hypercenter	C₈	D₃₁₅₇
hypocenter	C₇×C₁₁×C₄₁	D₈
LCS	G ▷ C₄×C₇×C₁₁×C₄₁ ▷ C₂×C₇×C₁₁×C₄₁ ▷ C₇×C₁₁×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 25256

A minimal faithful permutation representation — 2 generators on 67 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);(2,8)(3,7)(4,6)(10,15)(11,14)(12,13)(17,26)(18,25)(19,24)(20,23)(21,22)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2^4,7,11,41|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^10,g^a=g^40

Conjugacy classes & character table

12631 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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 7, 11, 41]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D25256 #72288435