C₈₉×(C₅⋊D₅)⋊C₃ #27211168

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

Aliases

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

Splittings

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

Subgroups

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

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 104
transitive degree	μ_t(G) = 1335

A minimal faithful permutation representation — 2 generators on 104 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	(C₅⋊D₅)⋊C₃×D₈₉, C₈₉×#64480, #40998920, …
maximal normal subgroup of	(C₅⋊D₅)⋊C₃×D₈₉, C₈₉×#64480, #40998920, …
maximal quotient of	C₈₉×#1039⋊C₃, C₂×C₈₉×(C₅⋊D₅)⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13350, 12);

C89×(C5⋊D5)⋊C3 #27211168