C₁₉×D₃₃₉ #27166263

order	12882
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×D₃₃₉
unambiguous semidirect product	C₃⋊(C₁₉×D₁₁₃)
unambiguous semidirect product	C₁₁₃⋊(D₃×C₁₉)

Splittings

Fitting, socle, Hall	C₃×C₁₉×C₁₁₃ ⋊ C₂
derived, Hall	C₃×C₁₁₃ ⋊ C₂×C₁₉
Hall	C₁₉×C₁₁₃ ⋊ D₃
Sylow	C₁₁₃ ⋊ D₃×C₁₉
Hall	C₃×C₁₉ ⋊ D₁₁₃
Sylow	C₃ ⋊ C₁₉×D₁₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉	D₃₃₉
commutator, cosocle, hypocenter	C₃×C₁₁₃	C₂×C₁₉
Frattini	C₁	G
Fitting, socle	C₃×C₁₉×C₁₁₃	C₂
LCS	G ▷ C₃×C₁₁₃
UCS	C₁ ◁ C₁₉

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

Conjugacy classes & character table

3249 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃×C₁₉×D₁₁₃, D₁₉×D₃₃₉, D₅₇⋊D₁₁₃, …
maximal normal subgroup of	D₃×C₁₉×D₁₁₃, D₁₉×D₃₃₉, D₅₇⋊D₁₁₃, …
maximal quotient of	C₁₉×Dic₃₃₉, C₂×C₁₉×D₃₃₉, C₁₉×D₁₀₁₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12882, 7);

C19×D339 #27166263