C₈₉×C₃⋊(C₅⋊D₅) #27211173

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

Aliases

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

Splittings

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

Subgroups

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

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 102
transitive degree	μ_t(G) = 6675

A minimal faithful permutation representation — 3 generators on 102 points:

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

Conjugacy classes & character table

3471 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅×D₁₅×C₈₉, D₄₄₅⋊D₁₅, C₃⋊(C₅⋊D₅)×D₈₉, …
maximal normal subgroup of	D₅×D₁₅×C₈₉, D₄₄₅⋊D₁₅, C₃⋊(C₅⋊D₅)×D₈₉, …
maximal quotient of	C₈₉×C₃⋊#1039, C₂×C₈₉×C₃⋊(C₅⋊D₅)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13350, 17);

C89×C3⋊(C5⋊D5) #27211173