D₉₁₃₉ #32912769

order	18278
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₁₃₉
unambiguous semidirect product	C₁₃⋊D₇₀₃
unambiguous semidirect product	C₁₉⋊D₄₈₁
unambiguous semidirect product	C₃₇⋊D₂₄₇

Splittings

Fitting, socle, derived, Hall	C₁₃×C₁₉×C₃₇ ⋊ C₂
Hall	C₁₉×C₃₇ ⋊ D₁₃
Hall	C₁₃×C₃₇ ⋊ D₁₉
Hall	C₁₃×C₁₉ ⋊ D₃₇
Sylow	C₃₇ ⋊ D₂₄₇
Sylow	C₁₉ ⋊ D₄₈₁
Sylow	C₁₃ ⋊ D₇₀₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₃×C₁₉×C₃₇	C₂
LCS	G ▷ C₁₃×C₁₉×C₃₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 9139

A minimal faithful permutation representation — 2 generators on 69 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);(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)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2,13,19,37|b^a=b^12,c^a=c^18,d^a=d^36

Conjugacy classes & character table

4571 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₇×D₂₄₇, D₁₉×D₄₈₁, D₁₃×D₇₀₃, …
maximal normal subgroup of	D₃₇×D₂₄₇, D₁₉×D₄₈₁, D₁₃×D₇₀₃, …
maximal quotient of	Dic₉₁₃₉, C₂×D₉₁₃₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18278, 8);

D9139 #32912769