D₁₅₈₂₇ #44539606

order	31654
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₈₂₇
unambiguous semidirect product	C₁₇⋊D₉₃₁
unambiguous semidirect product	C₁₉⋊D₈₃₃
unambiguous semidirect product	C₄₉⋊D₃₂₃

Splittings

Fitting, derived, Hall	C₁₇×C₁₉×C₄₉ ⋊ C₂
Hall	C₁₉×C₄₉ ⋊ D₁₇
Hall	C₁₇×C₄₉ ⋊ D₁₉
Hall	C₁₇×C₁₉ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₃₂₃
Sylow	C₁₉ ⋊ D₈₃₃
Sylow	C₁₇ ⋊ D₉₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₇×C₁₉×C₄₉	C₂
Frattini	C₇	D₂₂₆₁
socle	C₇×C₁₇×C₁₉	D₇
LCS	G ▷ C₁₇×C₁₉×C₄₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 15827

A minimal faithful permutation representation — 2 generators on 85 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);(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)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)

Polycyclic presentation:

2,7,17,19,7|b^7=e^6,b^a=b^6e,c^a=c^16,d^a=d^18,e^a=e^6

Conjugacy classes & character table

7915 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₈₂₇
maximal normal subgroup of	C₂×D₁₅₈₂₇
maximal quotient of	Dic₁₅₈₂₇, C₂×D₁₅₈₂₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31654, 8);

D15827 #44539606