C₃×C₁₃⋊C₃×C₁₁₃ #27210261

order	13221
isoclinism	isoclinic class of C₁₃⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃²×C₁₁₃ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×C₁₃⋊C₃×C₁₁₃
unambiguous semidirect product	C₁₃⋊(C₃²×C₁₁₃)
unambiguous semidirect product	(C₃×C₁₃)⋊(C₃×C₁₁₃)
unambiguous semidirect product	(C₁₃×C₁₁₃)⋊C₃²
unambiguous semidirect product	(C₃×C₁₃×C₁₁₃)⋊C₃

Splittings

Fitting, socle	C₃×C₁₃×C₁₁₃ ⋊ C₃
derived, Sylow	C₁₃ ⋊ C₃²×C₁₁₃
Hall	C₁₃×C₁₁₃ ⋊ C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₁₃	C₁₃⋊C₃
commutator, cosocle, hypocenter	C₁₃	C₃²×C₁₁₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₃×C₁₁₃	C₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃×C₁₁₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 4407

A minimal faithful permutation representation — 2 generators on 129 points:

(2,3,5)(4,7,10)(6,9,8)(11,13,12)(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);(1,2,4,3,6,10,7,11,13,5,8,12,9)(127,128,129)

Conjugacy classes & character table

2373 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×D₁₃⋊C₃×C₁₁₃, C₃×(C₁₃⋊C₃)⋊D₁₁₃, C₃×C₁₃⋊C₃×D₁₁₃, …
maximal normal subgroup of	C₃×D₁₃⋊C₃×C₁₁₃, C₃×(C₁₃⋊C₃)⋊D₁₁₃, C₃×C₁₃⋊C₃×D₁₁₃, …
maximal quotient of	C₂×C₃×C₁₃⋊C₃×C₁₁₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13221, 4);

C3×C13⋊C3×C113 #27210261