C₁₁₃×D₁₉⋊C₃ #27166267

order	12882
isoclinism	isoclinic class of D₁₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₁₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₁₃×D₁₉⋊C₃
unambiguous semidirect product	D₁₉⋊(C₃×C₁₁₃)
unambiguous semidirect product	(C₁₉⋊C₃)⋊(C₂×C₁₁₃)
unambiguous semidirect product	(D₁₉×C₁₁₃)⋊C₃

Splittings

Fitting, socle, Hall	C₁₉×C₁₁₃ ⋊ C₂×C₃
derived, Sylow	C₁₉ ⋊ C₂×C₃×C₁₁₃
Hall	C₁₉⋊C₃×C₁₁₃ ⋊ C₂
Hall	D₁₉×C₁₁₃ ⋊ C₃
Hall	C₁₉⋊C₃ ⋊ C₂×C₁₁₃
Hall	D₁₉ ⋊ C₃×C₁₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁₃	D₁₉⋊C₃
commutator, cosocle, hypocenter	C₁₉	C₂×C₃×C₁₁₃
Frattini	C₁	G
Fitting, socle	C₁₉×C₁₁₃	C₂×C₃
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₁₁₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 2147

A minimal faithful permutation representation — 2 generators on 132 points:

(2,6,7,3,4,8)(5,14,11,9,10,16)(12,17,19,18,15,13)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132);(1,2,5,11,17,14,18,4,8,13,19,7,6,12,10,15,16,9,3)

Conjugacy classes & character table

1017 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₉⋊C₃×D₁₁₃, C₂×C₁₁₃×D₁₉⋊C₃, C₁₁₃×AGL(1,𝔽₁₉), …
maximal normal subgroup of	D₁₉⋊C₃×D₁₁₃, C₂×C₁₁₃×D₁₉⋊C₃, C₁₁₃×AGL(1,𝔽₁₉), …
maximal quotient of	C₁₁₃×(C₁₉⋊C₃)⋊C₄, C₂×C₁₁₃×D₁₉⋊C₃, C₁₁₃×#4294⋊C₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12882, 11);

C113×D19⋊C3 #27166267