D₃×C₁₉×C₁₁₃ #27166262

order	12882
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₉×C₁₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₁₉×C₁₁₃
unambiguous semidirect product	C₃⋊(C₂×C₁₉×C₁₁₃)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉×C₁₁₃	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₁₉×C₁₁₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₉×C₁₁₃	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₁₉×C₁₁₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 6441

A minimal faithful permutation representation — 2 generators on 135 points:

(2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116)(117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(1,2,3)

Conjugacy classes & character table

6441 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃×C₁₉×D₁₁₃, D₃×D₁₉×C₁₁₃, D₃×D₂₁₄₇, …
maximal normal subgroup of	D₃×C₁₉×D₁₁₃, D₃×D₁₉×C₁₁₃, D₃×D₂₁₄₇, …
maximal quotient of	Dic₃×C₁₉×C₁₁₃, C₂×D₃×C₁₉×C₁₁₃, D₉×C₁₉×C₁₁₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12882, 6);

D3×C19×C113 #27166262