D₉₃₆₁ #34035366

order	18722
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₃₆₁
unambiguous semidirect product	C₁₁⋊D₈₅₁
unambiguous semidirect product	C₂₃⋊D₄₀₇
unambiguous semidirect product	C₃₇⋊D₂₅₃

Splittings

Fitting, socle, derived, Hall	C₁₁×C₂₃×C₃₇ ⋊ C₂
Hall	C₂₃×C₃₇ ⋊ D₁₁
Hall	C₁₁×C₃₇ ⋊ D₂₃
Hall	C₁₁×C₂₃ ⋊ D₃₇
Sylow	C₃₇ ⋊ D₂₅₃
Sylow	C₂₃ ⋊ D₄₀₇
Sylow	C₁₁ ⋊ D₈₅₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₁×C₂₃×C₃₇	C₂
LCS	G ▷ C₁₁×C₂₃×C₃₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 9361

A minimal faithful permutation representation — 2 generators on 71 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);(2,11)(3,10)(4,9)(5,8)(6,7)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Polycyclic presentation:

2,11,23,37|b^a=b^10,c^a=c^22,d^a=d^36

Conjugacy classes & character table

4682 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₇×D₂₅₃, D₂₃×D₄₀₇, D₁₁×D₈₅₁, …
maximal normal subgroup of	D₃₇×D₂₅₃, D₂₃×D₄₀₇, D₁₁×D₈₅₁, …
maximal quotient of	Dic₉₃₆₁, C₂×D₉₃₆₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18722, 9);

D9361 #34035366