D₅×C₁₂₁ #11925886

order	1210
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₂₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅×C₁₂₁
unambiguous semidirect product	C₅⋊(C₂×C₁₂₁)

Splittings

Fitting, Hall	C₅×C₁₂₁ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₁₂₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁₂₁	D₅
commutator, hypocenter	C₅	C₂×C₁₂₁
Frattini	C₁₁	D₅×C₁₁
Fitting	C₅×C₁₂₁	C₂
socle, cosocle	C₅×C₁₁	C₂×C₁₁
prime radical	D₅×C₁₁	C₁₁
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₂₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 126
transitive degree	μ_t(G) = 605

A minimal faithful permutation representation — 2 generators on 126 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);(2,5)(3,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	AGL(1,𝔽₅)×C₁₂₁, C₁₂₁⋊AGL(1,𝔽₅), D₅×D₁₂₁, …
maximal normal subgroup of	AGL(1,𝔽₅)×C₁₂₁, C₁₂₁⋊AGL(1,𝔽₅), D₅×D₁₂₁, …
maximal quotient of	Dic₅×C₁₂₁, C₂×D₅×C₁₂₁, C₃×D₅×C₁₂₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1210, 3);

D5×C121 #11925886