C₅×D₁₃₃₉ #27211481

order	13390
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₁₃₃₉
unambiguous semidirect product	C₁₃⋊(C₅×D₁₀₃)
unambiguous semidirect product	C₁₀₃⋊(C₅×D₁₃)

Splittings

Fitting, socle, Hall	C₅×C₁₃×C₁₀₃ ⋊ C₂
derived, Hall	C₁₃×C₁₀₃ ⋊ C₂×C₅
Hall	C₅×C₁₀₃ ⋊ D₁₃
Sylow	C₁₀₃ ⋊ C₅×D₁₃
Hall	C₅×C₁₃ ⋊ D₁₀₃
Sylow	C₁₃ ⋊ C₅×D₁₀₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₁₃₃₉
commutator, cosocle, hypocenter	C₁₃×C₁₀₃	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₁₃×C₁₀₃	C₂
LCS	G ▷ C₁₃×C₁₀₃
UCS	C₁ ◁ C₅

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 6695

A minimal faithful permutation representation — 2 generators on 121 points:

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

Conjugacy classes & character table

3355 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×D₁₃×D₁₀₃, D₅×D₁₃₃₉, D₆₅⋊D₁₀₃, …
maximal normal subgroup of	C₅×D₁₃×D₁₀₃, D₅×D₁₃₃₉, D₆₅⋊D₁₀₃, …
maximal quotient of	C₅×Dic₁₃₃₉, C₂×C₅×D₁₃₃₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13390, 4);

C5×D1339 #27211481