D₁₉×C₁₂₁ #17987148

order	4598
isoclinism	isoclinic class of D₁₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₂₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₁₉×C₁₂₁
unambiguous semidirect product	C₁₉⋊(C₂×C₁₂₁)

Splittings

Fitting, Hall	C₁₉×C₁₂₁ ⋊ C₂
derived, Sylow	C₁₉ ⋊ C₂×C₁₂₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁₂₁	D₁₉
commutator, hypocenter	C₁₉	C₂×C₁₂₁
Frattini	C₁₁	C₁₁×D₁₉
Fitting	C₁₉×C₁₂₁	C₂
socle, cosocle	C₁₁×C₁₉	C₂×C₁₁
prime radical	C₁₁×D₁₉	C₁₁
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₁₂₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 140
transitive degree	μ_t(G) = 2299

A minimal faithful permutation representation — 2 generators on 140 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140);(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)

Conjugacy classes & character table

1331 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₉⋊C₃×C₁₂₁, C₃×D₁₉×C₁₂₁, D₅₇×C₁₂₁, …
maximal normal subgroup of	D₁₉⋊C₃×C₁₂₁, C₃×D₁₉×C₁₂₁, C₅×D₁₉×C₁₂₁, …
maximal quotient of	C₃×D₁₉×C₁₂₁, D₅₇×C₁₂₁, C₅×D₁₉×C₁₂₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4598, 3);

D19×C121 #17987148