C₃×D₅×C₁₂₁ #16813787

order	3630
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₁₂₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₅×C₁₂₁
unambiguous semidirect product	C₅⋊(C₂×C₃×C₁₂₁)

Splittings

Fitting, Hall	C₃×C₅×C₁₂₁ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₃×C₁₂₁
Hall	C₅×C₁₂₁ ⋊ C₂×C₃
Hall	C₃×C₅ ⋊ C₂×C₁₂₁

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₂₁	D₅
commutator, hypocenter	C₅	C₂×C₃×C₁₂₁
Frattini	C₁₁	C₃×D₅×C₁₁
Fitting	C₃×C₅×C₁₂₁	C₂
socle	C₃×C₅×C₁₁	C₂×C₁₁
prime radical	C₃×D₅×C₁₁	C₁₁
cosocle	C₅×C₁₁	C₂×C₃×C₁₁
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₁₂₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 1815

A minimal faithful permutation representation — 2 generators on 129 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);(5,8)(6,7)

Conjugacy classes & character table

1452 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉×D₅×C₁₂₁, C₃²×D₅×C₁₂₁, C₃×D₁₅×C₁₂₁, …
maximal normal subgroup of	C₉×D₅×C₁₂₁, C₃²×D₅×C₁₂₁, C₃×D₅×C₁₂₁⋊C₅, …
maximal quotient of	C₉×D₅×C₁₂₁, C₃²×D₅×C₁₂₁, C₃×D₁₅×C₁₂₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3630, 7);

C3×D5×C121 #16813787