D₇₆₅₇ #29506148

order	15314
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₆₅₇
unambiguous semidirect product	C₁₃⋊D₅₈₉
unambiguous semidirect product	C₁₉⋊D₄₀₃
unambiguous semidirect product	C₃₁⋊D₂₄₇

Splittings

Fitting, socle, derived, Hall	C₁₃×C₁₉×C₃₁ ⋊ C₂
Hall	C₁₉×C₃₁ ⋊ D₁₃
Hall	C₁₃×C₃₁ ⋊ D₁₉
Hall	C₁₃×C₁₉ ⋊ D₃₁
Sylow	C₃₁ ⋊ D₂₄₇
Sylow	C₁₉ ⋊ D₄₀₃
Sylow	C₁₃ ⋊ D₅₈₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₃×C₁₉×C₃₁	C₂
LCS	G ▷ C₁₃×C₁₉×C₃₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 7657

A minimal faithful permutation representation — 2 generators on 63 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);(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2,13,19,31|b^a=b^12,c^a=c^18,d^a=d^30

Conjugacy classes & character table

3830 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁×D₂₄₇, D₁₉×D₄₀₃, D₁₃×D₅₈₉, …
maximal normal subgroup of	D₃₁×D₂₄₇, D₁₉×D₄₀₃, D₁₃×D₅₈₉, …
maximal quotient of	Dic₇₆₅₇, C₂×D₇₆₅₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15314, 8);

D7657 #29506148