C₃₁×D₂₀₉ #27188372

order	12958
isoclinism	isoclinic class of D₂₀₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃₁×D₂₀₉
unambiguous semidirect product	C₁₁⋊(C₃₁×D₁₉)
unambiguous semidirect product	C₁₉⋊(D₁₁×C₃₁)

Splittings

Fitting, socle, Hall	C₁₁×C₁₉×C₃₁ ⋊ C₂
derived, Hall	C₁₁×C₁₉ ⋊ C₂×C₃₁
Hall	C₁₉×C₃₁ ⋊ D₁₁
Hall	C₁₁×C₃₁ ⋊ D₁₉
Sylow	C₁₉ ⋊ D₁₁×C₃₁
Sylow	C₁₁ ⋊ C₃₁×D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃₁	D₂₀₉
commutator, cosocle, hypocenter	C₁₁×C₁₉	C₂×C₃₁
Frattini	C₁	G
Fitting, socle	C₁₁×C₁₉×C₃₁	C₂
LCS	G ▷ C₁₁×C₁₉
UCS	C₁ ◁ C₃₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 6479

A minimal faithful permutation representation — 2 generators on 61 points:

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

Conjugacy classes & character table

3286 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁×D₂₀₉, D₁₁×C₃₁×D₁₉, D₂₀₉⋊D₃₁, …
maximal normal subgroup of	D₃₁×D₂₀₉, D₁₁×C₃₁×D₁₉, D₂₀₉⋊D₃₁, …
maximal quotient of	C₃₁×Dic₂₀₉, C₂×C₃₁×D₂₀₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12958, 7);

C31×D209 #27188372