C₁₁₃×C₁₃⋊C₉ #27210259

order	13221
isoclinism	isoclinic class of C₁₃⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₉×C₁₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₁₃×C₁₃⋊C₉
unambiguous semidirect product	C₁₃⋊(C₉×C₁₁₃)
unambiguous semidirect product	(C₁₃×C₁₁₃)⋊C₉

Splittings

derived, Sylow	C₁₃ ⋊ C₉×C₁₁₃
Hall	C₁₃×C₁₁₃ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₁₃	C₁₃⋊C₃
commutator, hypocenter	C₁₃	C₉×C₁₁₃
Frattini	C₃	C₁₃⋊C₃×C₁₁₃
Fitting, socle, prime radical	C₃×C₁₃×C₁₁₃	C₃
cosocle	C₃×C₁₃	C₃×C₁₁₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃×C₁₁₃

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 13221

A minimal faithful permutation representation — 2 generators on 135 points:

(2,3,5)(4,7,10)(6,9,8)(11,13,12)(14,15,17,16,18,20,19,21,22)(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);(1,2,4,3,6,10,7,11,13,5,8,12,9)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₁₃×C₁₃⋊C₉, C₁₃⋊C₉×D₁₁₃, C₁₁₃×D₁₃⋊C₉, …
maximal normal subgroup of	C₂×C₁₁₃×C₁₃⋊C₉, C₁₃⋊C₉×D₁₁₃, C₁₁₃×D₁₃⋊C₉, …
maximal quotient of	C₂×C₁₁₃×C₁₃⋊C₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13221, 2);

C113×C13⋊C9 #27210259