D₁₀₉₆₅ #36346704

order	21930
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₉₆₅
unambiguous semidirect product	C₃⋊D₃₆₅₅
unambiguous semidirect product	C₅⋊D₂₁₉₃
unambiguous semidirect product	C₁₇⋊D₆₄₅
unambiguous semidirect product	C₄₃⋊D₂₅₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₁₇×C₄₃ ⋊ C₂
Hall	C₅×C₁₇×C₄₃ ⋊ D₃
Hall	C₃×C₁₇×C₄₃ ⋊ D₅
Hall	C₁₇×C₄₃ ⋊ D₁₅
Hall	C₃×C₅×C₄₃ ⋊ D₁₇
Hall	C₃×C₅×C₁₇ ⋊ D₄₃
Hall	C₅×C₄₃ ⋊ D₅₁
Hall	C₃×C₄₃ ⋊ D₈₅
Hall	C₅×C₁₇ ⋊ D₁₂₉
Hall	C₃×C₁₇ ⋊ D₂₁₅
Sylow	C₄₃ ⋊ D₂₅₅
Sylow	C₁₇ ⋊ D₆₄₅
Hall	C₃×C₅ ⋊ D₇₃₁
Sylow	C₅ ⋊ D₂₁₉₃
Sylow	C₃ ⋊ D₃₆₅₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₁₇×C₄₃	C₂
LCS	G ▷ C₃×C₅×C₁₇×C₄₃
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 10965

A minimal faithful permutation representation — 2 generators on 68 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);(2,3)(5,8)(6,7)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2,3,5,17,43|b^a=b^2,c^a=c^4,d^a=d^16,e^a=e^42

Conjugacy classes & character table

5484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21930, 17);

D10965 #36346704