D₁₆₂₁₅ #44569870

order	32430
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₂₁₅
unambiguous semidirect product	C₃⋊D₅₄₀₅
unambiguous semidirect product	C₅⋊D₃₂₄₃
unambiguous semidirect product	C₂₃⋊D₇₀₅
unambiguous semidirect product	C₄₇⋊D₃₄₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₂₃×C₄₇ ⋊ C₂
Hall	C₅×C₂₃×C₄₇ ⋊ D₃
Hall	C₃×C₂₃×C₄₇ ⋊ D₅
Hall	C₂₃×C₄₇ ⋊ D₁₅
Hall	C₃×C₅×C₄₇ ⋊ D₂₃
Hall	C₃×C₅×C₂₃ ⋊ D₄₇
Hall	C₅×C₄₇ ⋊ D₆₉
Hall	C₃×C₄₇ ⋊ D₁₁₅
Hall	C₅×C₂₃ ⋊ D₁₄₁
Hall	C₃×C₂₃ ⋊ D₂₃₅
Sylow	C₄₇ ⋊ D₃₄₅
Sylow	C₂₃ ⋊ D₇₀₅
Hall	C₃×C₅ ⋊ D₁₀₈₁
Sylow	C₅ ⋊ D₃₂₄₃
Sylow	C₃ ⋊ D₅₄₀₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₂₃×C₄₇	C₂
LCS	G ▷ C₃×C₅×C₂₃×C₄₇
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 78
transitive degree	μ_t(G) = 16215

A minimal faithful permutation representation — 2 generators on 78 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);(2,3)(5,8)(6,7)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2,3,5,23,47|b^a=b^2,c^a=c^4,d^a=d^22,e^a=e^46

Conjugacy classes & character table

8109 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₆₂₁₅
maximal normal subgroup of	C₂×D₁₆₂₁₅
maximal quotient of	Dic₁₆₂₁₅, C₂×D₁₆₂₁₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32430, 17);

D16215 #44569870