C₅₉×C₅²⋊C₉ #27210758

order	13275
isoclinism	isoclinic class of C₅²⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₉×C₅₉ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅₉×C₅²⋊C₉
unambiguous semidirect product	C₅²⋊(C₉×C₅₉)
unambiguous semidirect product	(C₅²×C₅₉)⋊C₉

Splittings

derived, Sylow	C₅² ⋊ C₉×C₅₉
Hall	C₅²×C₅₉ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅₉	C₅²⋊C₃
commutator, hypocenter	C₅²	C₉×C₅₉
Frattini	C₃	C₅₉×C₅²⋊C₃
Fitting, socle, prime radical	C₃×C₅²×C₅₉	C₃
cosocle	C₃×C₅²	C₃×C₅₉
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃×C₅₉

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 2655

A minimal faithful permutation representation — 2 generators on 83 points:

(1,2,4)(3,6,8)(5,10,13)(7,12,14)(9,15,11)(16,17,19,18,20,22,21,23,24)(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);(1,3,7,13,15)(2,6,12,5,11)(4,10,8,9,14)

Conjugacy classes & character table

1947 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	C₅₉⋊#90646
maximal subgroup of	C₅₉×#90646, C₅₉⋊#90646, C₅₉×(C₅⋊D₅)⋊C₉, …
maximal normal subgroup of	C₅₉×#90646, C₅₉⋊#90646, C₅₉×(C₅⋊D₅)⋊C₉, …
maximal quotient of	C₂×C₅₉×C₅²⋊C₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13275, 4);

C59×C52⋊C9 #27210758