C₃×C₅×D₅×C₈₉ #27211181

order	13350
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅×C₈₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₅×D₅×C₈₉
unambiguous semidirect product	C₅⋊(C₂×C₃×C₅×C₈₉)

Splittings

Fitting, socle, Hall	C₃×C₅²×C₈₉ ⋊ C₂
derived	C₅ ⋊ C₂×C₃×C₅×C₈₉
Hall	C₅²×C₈₉ ⋊ C₂×C₃
Hall	C₃×C₅² ⋊ C₂×C₈₉
Sylow	C₅² ⋊ C₂×C₃×C₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅×C₈₉	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₃×C₅×C₈₉
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₈₉	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₅×C₈₉

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 102
transitive degree	μ_t(G) = 2670

A minimal faithful permutation representation — 2 generators on 102 points:

(1,2,3,4,5)(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);(6,7,9,10,8)

Conjugacy classes & character table

5340 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₅×D₅×D₈₉, C₃×C₅×AGL(1,𝔽₅)×C₈₉, C₃×C₅×#13184069, …
maximal normal subgroup of	C₃×C₅×D₅×D₈₉, C₃×C₅×AGL(1,𝔽₅)×C₈₉, C₃×C₅×#13184069, …
maximal quotient of	C₃×C₅×Dic₅×C₈₉, C₂×C₃×C₅×D₅×C₈₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13350, 25);

C3×C5×D5×C89 #27211181