C₃×C₁₃⋊D₂₉₉ #37552246

order	23322
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁₃

Aliases

direct product	C₃×C₁₃⋊D₂₉₉
unambiguous semidirect product	C₂₃⋊(C₃×C₁₃⋊D₁₃)

Splittings

Fitting, socle, Hall	C₃×C₁₃²×C₂₃ ⋊ C₂
derived, Hall	C₁₃²×C₂₃ ⋊ C₂×C₃
Hall	C₃×C₁₃² ⋊ D₂₃
Sylow	C₁₃² ⋊ C₃×D₂₃
Hall	C₃×C₂₃ ⋊ C₁₃⋊D₁₃
Sylow	C₂₃ ⋊ C₃×C₁₃⋊D₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₁₃⋊D₂₉₉
commutator, cosocle, hypocenter	C₁₃²×C₂₃	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₃²×C₂₃	C₂
LCS	G ▷ C₁₃²×C₂₃
UCS	C₁ ◁ C₃

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 11661

A minimal faithful permutation representation — 3 generators on 52 points:

(1,2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52);(4,5,7,9,11,13,15,17,19,21,23,25,26,24,22,20,18,16,14,12,10,8,6)(40,41,43,45,47,49,51,52,50,48,46,44,42);(27,28,30,32,34,36,38,39,37,35,33,31,29)

Conjugacy classes & character table

5835 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23322, 40);

C3×C13⋊D299 #37552246