D₁₅₆₃₁ #44530493

order	31262
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₆₃₁
unambiguous semidirect product	C₁₁⋊D₁₄₂₁
unambiguous semidirect product	C₂₉⋊D₅₃₉
unambiguous semidirect product	C₄₉⋊D₃₁₉

Splittings

Fitting, derived, Hall	C₁₁×C₂₉×C₄₉ ⋊ C₂
Hall	C₂₉×C₄₉ ⋊ D₁₁
Hall	C₁₁×C₄₉ ⋊ D₂₉
Hall	C₁₁×C₂₉ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₃₁₉
Sylow	C₂₉ ⋊ D₅₃₉
Sylow	C₁₁ ⋊ D₁₄₂₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₁×C₂₉×C₄₉	C₂
Frattini	C₇	D₂₂₃₃
socle	C₇×C₁₁×C₂₉	D₇
LCS	G ▷ C₁₁×C₂₉×C₄₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 15631

A minimal faithful permutation representation — 2 generators on 89 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);(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)(51,60)(52,59)(53,58)(54,57)(55,56)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2,7,11,29,7|b^7=e^6,b^a=b^6e,c^a=c^10,d^a=d^28,e^a=e^6

Conjugacy classes & character table

7817 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₆₃₁
maximal normal subgroup of	C₂×D₁₅₆₃₁
maximal quotient of	Dic₁₅₆₃₁, C₂×D₁₅₆₃₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31262, 9);

D15631 #44530493