D₁₅₈₄₇ #44540136

order	31694
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₈₄₇
unambiguous semidirect product	C₁₃⋊D₁₂₁₉
unambiguous semidirect product	C₂₃⋊D₆₈₉
unambiguous semidirect product	C₅₃⋊D₂₉₉

Splittings

Fitting, socle, derived, Hall	C₁₃×C₂₃×C₅₃ ⋊ C₂
Hall	C₂₃×C₅₃ ⋊ D₁₃
Hall	C₁₃×C₅₃ ⋊ D₂₃
Hall	C₁₃×C₂₃ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₂₉₉
Sylow	C₂₃ ⋊ D₆₈₉
Sylow	C₁₃ ⋊ D₁₂₁₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₃×C₂₃×C₅₃	C₂
LCS	G ▷ C₁₃×C₂₃×C₅₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 15847

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);(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2,13,23,53|b^a=b^12,c^a=c^22,d^a=d^52

Conjugacy classes & character table

7925 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₈₄₇
maximal normal subgroup of	C₂×D₁₅₈₄₇
maximal quotient of	Dic₁₅₈₄₇, C₂×D₁₅₈₄₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31694, 9);

D15847 #44540136