C₅×D₁₅×C₈₉ #27211175

order	13350
isoclinism	isoclinic class of D₁₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₈₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₁₅×C₈₉
unambiguous semidirect product	C₃⋊(C₅×D₅×C₈₉)
unambiguous semidirect product	C₅⋊(C₅×D₃×C₈₉)

Splittings

Fitting, socle, Hall	C₃×C₅²×C₈₉ ⋊ C₂
derived	C₃×C₅ ⋊ C₂×C₅×C₈₉
Hall	C₅²×C₈₉ ⋊ D₃
Hall	C₃×C₈₉ ⋊ C₅×D₅
Hall	C₃×C₅² ⋊ C₂×C₈₉
Sylow	C₅² ⋊ D₃×C₈₉
Sylow	C₃ ⋊ C₅×D₅×C₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₈₉	D₁₅
commutator, cosocle, hypocenter	C₃×C₅	C₂×C₅×C₈₉
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₈₉	C₂
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ 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)(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)(11,12,13)

Conjugacy classes & character table

4005 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×D₁₅×D₈₉, C₅×D₃×D₅×C₈₉, C₅×D₁₅⋊D₈₉, …
maximal normal subgroup of	C₅×D₁₅×D₈₉, C₅×D₃×D₅×C₈₉, C₅×D₁₅⋊D₈₉, …
maximal quotient of	C₅×Dic₁₅×C₈₉, C₂×C₅×D₁₅×C₈₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13350, 19);

C5×D15×C89 #27211175