D₁₁₀₇₇ #36370640

order	22154
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₀₇₇
unambiguous semidirect product	C₁₁⋊D₁₀₀₇
unambiguous semidirect product	C₁₉⋊D₅₈₃
unambiguous semidirect product	C₅₃⋊D₂₀₉

Splittings

Fitting, socle, derived, Hall	C₁₁×C₁₉×C₅₃ ⋊ C₂
Hall	C₁₉×C₅₃ ⋊ D₁₁
Hall	C₁₁×C₅₃ ⋊ D₁₉
Hall	C₁₁×C₁₉ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₂₀₉
Sylow	C₁₉ ⋊ D₅₈₃
Sylow	C₁₁ ⋊ D₁₀₀₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₁×C₁₉×C₅₃	C₂
LCS	G ▷ C₁₁×C₁₉×C₅₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 11077

A minimal faithful permutation representation — 2 generators on 83 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);(2,11)(3,10)(4,9)(5,8)(6,7)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(32,83)(33,82)(34,81)(35,80)(36,79)(37,78)(38,77)(39,76)(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:

2,11,19,53|b^a=b^10,c^a=c^18,d^a=d^52

Conjugacy classes & character table

5540 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22154, 8);

D11077 #36370640