C₂×C₁₉⋊C₃×C₁₁₃ #27166258

order	12882
isoclinism	isoclinic class of C₁₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₁₁₃ · Schur multiplier (H₂) = C₁

Aliases

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

Splittings

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

Subgroups

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

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 134
transitive degree	μ_t(G) = 4294

A minimal faithful permutation representation — 2 generators on 134 points:

(2,3,5)(4,7,8)(6,10,9)(11,15,13)(12,16,14)(17,19,18)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132)(133,134);(1,2,4,8,13,18,15,3,6,11,16,5,9,14,7,12,17,19,10)

Conjugacy classes & character table

2034 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄×C₁₉⋊C₃×C₁₁₃, C₁₉⋊C₃×Dic₁₁₃, C₁₁₃×(C₁₉⋊C₃)⋊C₄, …
maximal normal subgroup of	C₄×C₁₉⋊C₃×C₁₁₃, C₁₉⋊C₃×Dic₁₁₃, C₁₁₃×(C₁₉⋊C₃)⋊C₄, …
maximal quotient of	C₄×C₁₉⋊C₃×C₁₁₃, C₂²×C₁₉⋊C₃×C₁₁₃, C₂×C₁₁₃×#4294, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12882, 2);

C2×C19⋊C3×C113 #27166258