C₁₃⋊C₃×C₅₃ #14523374

order	2067
isoclinism	isoclinic class of C₁₃⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃⋊C₃×C₅₃
unambiguous semidirect product	C₁₃⋊(C₃×C₅₃)
unambiguous semidirect product	(C₁₃×C₅₃)⋊C₃

Splittings

Fitting, socle, Hall	C₁₃×C₅₃ ⋊ C₃
derived, Sylow	C₁₃ ⋊ C₃×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	C₁₃⋊C₃
commutator, cosocle, hypocenter	C₁₃	C₃×C₅₃
Frattini	C₁	G
Fitting, socle	C₁₃×C₅₃	C₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₅₃

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 689

A minimal faithful permutation representation — 2 generators on 66 points:

(2,3,5)(4,7,10)(6,9,8)(11,13,12)(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);(1,2,4,3,6,10,7,11,13,5,8,12,9)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₃⋊C₃×C₅₃, C₁₃⋊C₃×D₅₃, C₅₃×D₁₃⋊C₃, …
maximal normal subgroup of	C₂×C₁₃⋊C₃×C₅₃, C₁₃⋊C₃×D₅₃, C₅₃×D₁₃⋊C₃, …
maximal quotient of	C₂×C₁₃⋊C₃×C₅₃, C₇×C₁₃⋊C₃×C₅₃, C₅₃×#63270, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2067, 2);

C13⋊C3×C53 #14523374