D₁₆₁₀₇ #44567266

order	32214
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₁₀₇
unambiguous semidirect product	C₃⋊D₅₃₆₉
unambiguous semidirect product	C₇⋊D₂₃₀₁
unambiguous semidirect product	C₁₃⋊D₁₂₃₉
unambiguous semidirect product	C₅₉⋊D₂₇₃

Splittings

Fitting, socle, derived, Hall	C₃×C₇×C₁₃×C₅₉ ⋊ C₂
Hall	C₇×C₁₃×C₅₉ ⋊ D₃
Hall	C₃×C₁₃×C₅₉ ⋊ D₇
Hall	C₃×C₇×C₅₉ ⋊ D₁₃
Hall	C₁₃×C₅₉ ⋊ D₂₁
Hall	C₇×C₅₉ ⋊ D₃₉
Hall	C₃×C₇×C₁₃ ⋊ D₅₉
Hall	C₃×C₅₉ ⋊ D₉₁
Hall	C₇×C₁₃ ⋊ D₁₇₇
Sylow	C₅₉ ⋊ D₂₇₃
Hall	C₃×C₁₃ ⋊ D₄₁₃
Hall	C₃×C₇ ⋊ D₇₆₇
Sylow	C₁₃ ⋊ D₁₂₃₉
Sylow	C₇ ⋊ D₂₃₀₁
Sylow	C₃ ⋊ D₅₃₆₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₃×C₅₉	C₂
LCS	G ▷ C₃×C₇×C₁₃×C₅₉
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 82
transitive degree	μ_t(G) = 16107

A minimal faithful permutation representation — 2 generators on 82 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);(2,3)(5,10)(6,9)(7,8)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(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:

2,3,7,13,59|b^a=b^2,c^a=c^6,d^a=d^12,e^a=e^58

Conjugacy classes & character table

8055 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₆₁₀₇
maximal normal subgroup of	C₂×D₁₆₁₀₇
maximal quotient of	Dic₁₆₁₀₇, C₂×D₁₆₁₀₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32214, 20);

D16107 #44567266