D₉₆₅₇ #34064537

order	19314
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₆₅₇
unambiguous semidirect product	C₉⋊D₁₀₇₃
unambiguous semidirect product	C₂₉⋊D₃₃₃
unambiguous semidirect product	C₃₇⋊D₂₆₁

Splittings

Fitting, derived, Hall	C₉×C₂₉×C₃₇ ⋊ C₂
Hall	C₂₉×C₃₇ ⋊ D₉
Hall	C₉×C₃₇ ⋊ D₂₉
Hall	C₉×C₂₉ ⋊ D₃₇
Sylow	C₃₇ ⋊ D₂₆₁
Sylow	C₂₉ ⋊ D₃₃₃
Sylow	C₉ ⋊ D₁₀₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₂₉×C₃₇	C₂
Frattini	C₃	D₃₂₁₉
socle	C₃×C₂₉×C₃₇	D₃
LCS	G ▷ C₉×C₂₉×C₃₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 9657

A minimal faithful permutation representation — 2 generators on 75 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);(2,9)(3,8)(4,7)(5,6)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)

Polycyclic presentation:

2,3,29,37,3|b^3=e^2,b^a=b^2e,c^a=c^28,d^a=d^36,e^a=e^2

Conjugacy classes & character table

4830 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₇×D₂₆₁, D₂₉×D₃₃₃, D₉×D₁₀₇₃, …
maximal normal subgroup of	D₃₇×D₂₆₁, D₂₉×D₃₃₃, D₉×D₁₀₇₃, …
maximal quotient of	Dic₉₆₅₇, C₂×D₉₆₅₇, D₂₈₉₇₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19314, 9);

D9657 #34064537