D₈₂₆₅ #30658356

order	16530
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₈₂₆₅
unambiguous semidirect product	C₃⋊D₂₇₅₅
unambiguous semidirect product	C₅⋊D₁₆₅₃
unambiguous semidirect product	C₁₉⋊D₄₃₅
unambiguous semidirect product	C₂₉⋊D₂₈₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₁₉×C₂₉ ⋊ C₂
Hall	C₅×C₁₉×C₂₉ ⋊ D₃
Hall	C₃×C₁₉×C₂₉ ⋊ D₅
Hall	C₁₉×C₂₉ ⋊ D₁₅
Hall	C₃×C₅×C₂₉ ⋊ D₁₉
Hall	C₃×C₅×C₁₉ ⋊ D₂₉
Hall	C₅×C₂₉ ⋊ D₅₇
Hall	C₅×C₁₉ ⋊ D₈₇
Hall	C₃×C₂₉ ⋊ D₉₅
Hall	C₃×C₁₉ ⋊ D₁₄₅
Sylow	C₂₉ ⋊ D₂₈₅
Sylow	C₁₉ ⋊ D₄₃₅
Hall	C₃×C₅ ⋊ D₅₅₁
Sylow	C₅ ⋊ D₁₆₅₃
Sylow	C₃ ⋊ D₂₇₅₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₁₉×C₂₉	C₂
LCS	G ▷ C₃×C₅×C₁₉×C₂₉
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 8265

A minimal faithful permutation representation — 2 generators on 56 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);(2,3)(5,8)(6,7)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)

Polycyclic presentation:

2,3,5,19,29|b^a=b^2,c^a=c^4,d^a=d^18,e^a=e^28

Conjugacy classes & character table

4134 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₉×D₂₈₅, D₁₉×D₄₃₅, D₁₅×D₅₅₁, …
maximal normal subgroup of	D₂₉×D₂₈₅, D₁₉×D₄₃₅, D₁₅×D₅₅₁, …
maximal quotient of	Dic₈₂₆₅, C₂×D₈₂₆₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16530, 17);

D8265 #30658356