C₅×C₁₁⋊D₂₀₉ #37527398

order	22990
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁₁

Aliases

direct product	C₅×C₁₁⋊D₂₀₉
unambiguous semidirect product	C₁₉⋊(C₅×C₁₁⋊D₁₁)

Splittings

Fitting, socle, Hall	C₅×C₁₁²×C₁₉ ⋊ C₂
derived, Hall	C₁₁²×C₁₉ ⋊ C₂×C₅
Hall	C₅×C₁₁² ⋊ D₁₉
Sylow	C₁₁² ⋊ C₅×D₁₉
Hall	C₅×C₁₉ ⋊ C₁₁⋊D₁₁
Sylow	C₁₉ ⋊ C₅×C₁₁⋊D₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₁₁⋊D₂₀₉
commutator, cosocle, hypocenter	C₁₁²×C₁₉	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₁₁²×C₁₉	C₂
LCS	G ▷ C₁₁²×C₁₉
UCS	C₁ ◁ C₅

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 46
transitive degree	μ_t(G) = 11495

A minimal faithful permutation representation — 3 generators on 46 points:

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

Conjugacy classes & character table

5755 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22990, 48);

C5×C11⋊D209 #37527398