C₅×C₂₃×C₁₃⋊C₉ #27212136

order	13455
isoclinism	isoclinic class of C₁₃⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₅×C₉×C₂₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₂₃×C₁₃⋊C₉
unambiguous semidirect product	C₁₃⋊(C₅×C₉×C₂₃)
unambiguous semidirect product	(C₅×C₁₃)⋊(C₉×C₂₃)
unambiguous semidirect product	(C₁₃×C₂₃)⋊(C₅×C₉)
unambiguous semidirect product	(C₅×C₁₃×C₂₃)⋊C₉

Splittings

derived, Sylow	C₁₃ ⋊ C₅×C₉×C₂₃
Hall	C₅×C₁₃×C₂₃ ⋊ C₉
Hall	C₁₃×C₂₃ ⋊ C₅×C₉
Hall	C₅×C₁₃ ⋊ C₉×C₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅×C₂₃	C₁₃⋊C₃
commutator, hypocenter	C₁₃	C₅×C₉×C₂₃
Frattini	C₃	C₅×C₂₃×C₁₃⋊C₃
Fitting, socle, prime radical	C₃×C₅×C₁₃×C₂₃	C₃
cosocle	C₃×C₁₃	C₃×C₅×C₂₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃×C₅×C₂₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 13455

A minimal faithful permutation representation — 2 generators on 50 points:

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

Conjugacy classes & character table

2415 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₅×C₂₃×C₁₃⋊C₉, C₅×D₂₃×C₁₃⋊C₉, C₅×C₂₃×D₁₃⋊C₉, …
maximal normal subgroup of	C₂×C₅×C₂₃×C₁₃⋊C₉, C₅×D₂₃×C₁₃⋊C₉, C₅×C₂₃×D₁₃⋊C₉, …
maximal quotient of	C₂×C₅×C₂₃×C₁₃⋊C₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13455, 2);

C5×C23×C13⋊C9 #27212136