C₂×C₁₉×C₃⋊(C₁₃⋊D₁₃) #50310541

order	38532
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₉ · Schur multiplier (H₂) = C₂×C₁₃

Aliases

direct product	C₂×C₁₉×C₃⋊(C₁₃⋊D₁₃)
unambiguous semidirect product	(C₂×C₃)⋊(C₁₉×C₁₃⋊D₁₃)
unambiguous semidirect product	(C₂×C₁₃)⋊(C₁₉×D₃₉)

Splittings

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₉	C₃⋊(C₁₃⋊D₁₃)
commutator, cosocle, hypocenter	C₃×C₁₃²	C₂²×C₁₉
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₃²×C₁₉	C₂
LCS	G ▷ C₃×C₁₃²
UCS	C₁ ◁ C₂×C₁₉

320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 19266

A minimal faithful permutation representation — 3 generators on 50 points:

(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(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);(1,2)(3,4,5)(19,20,22,24,26,28,30,31,29,27,25,23,21);(6,7,9,11,13,15,17,18,16,14,12,10,8)

Conjugacy classes & character table

9690 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38532, 423);

Fitting, socle	C₂×C₃×C₁₃²×C₁₉ ⋊ C₂
derived, Hall	C₃×C₁₃² ⋊ C₂²×C₁₉
Hall	C₃×C₁₃²×C₁₉ ⋊ C₂²
Hall	C₁₃²×C₁₉ ⋊ C₂×D₃
Sylow	C₁₃² ⋊ C₂×D₃×C₁₉
Hall	C₃×C₁₉ ⋊ C₂×C₁₃⋊D₁₃
Sylow	C₃ ⋊ C₂×C₁₉×C₁₃⋊D₁₃

C2×C19×C3⋊(C13⋊D13) #50310541