D₁₅×C₅₃ #13097553

order	1590
isoclinism	isoclinic class of D₁₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₁₅×C₅₃
unambiguous semidirect product	C₃⋊(D₅×C₅₃)
unambiguous semidirect product	C₅⋊(D₃×C₅₃)

Splittings

Fitting, socle, Hall	C₃×C₅×C₅₃ ⋊ C₂
derived, Hall	C₃×C₅ ⋊ C₂×C₅₃
Hall	C₅×C₅₃ ⋊ D₃
Hall	C₃×C₅₃ ⋊ D₅
Sylow	C₅ ⋊ D₃×C₅₃
Sylow	C₃ ⋊ D₅×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	D₁₅
commutator, cosocle, hypocenter	C₃×C₅	C₂×C₅₃
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₅₃	C₂
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₅₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 795

A minimal faithful permutation representation — 2 generators on 61 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);(2,3)(5,8)(6,7)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₁₅×D₅₃, D₃×D₅×C₅₃, D₁₅⋊D₅₃, …
maximal normal subgroup of	D₁₅×D₅₃, D₃×D₅×C₅₃, D₁₅⋊D₅₃, …
maximal quotient of	C₅₃×Dic₁₅, C₂×D₁₅×C₅₃, C₅₃×D₄₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1590, 6);

D15×C53 #13097553