C₅×C₁₃×C₄₁⋊C₅ #27211019

order	13325
isoclinism	isoclinic class of C₄₁⋊C₅
automorphism	?
homology	Abelianization (H₁) = C₅²×C₁₃ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅×C₁₃×C₄₁⋊C₅
unambiguous semidirect product	C₄₁⋊(C₅²×C₁₃)
unambiguous semidirect product	(C₅×C₄₁)⋊(C₅×C₁₃)
unambiguous semidirect product	(C₁₃×C₄₁)⋊C₅²
unambiguous semidirect product	(C₅×C₁₃×C₄₁)⋊C₅

Splittings

Fitting, socle	C₅×C₁₃×C₄₁ ⋊ C₅
derived, Sylow	C₄₁ ⋊ C₅²×C₁₃
Hall	C₁₃×C₄₁ ⋊ C₅²

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₁₃	C₄₁⋊C₅
commutator, cosocle, hypocenter	C₄₁	C₅²×C₁₃
Frattini	C₁	G
Fitting, socle	C₅×C₁₃×C₄₁	C₅
LCS	G ▷ C₄₁
UCS	C₁ ◁ C₅×C₁₃

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 2665

A minimal faithful permutation representation — 2 generators on 59 points:

(2,3,5,9,17)(4,7,13,24,35)(6,11,20,12,22)(8,15,23,34,21)(10,19,30,18,29)(14,25,36,40,33)(16,28,27,38,41)(26,37,31,39,32)(42,43,44,45,46,47,48,49,50,51,52,53,54);(1,2,4,8,16,25,29,34,40,36,3,6,12,23,30,37,9,18,5,10,7,14,26,38,39,41,19,31,11,21,15,27,24,35,20,32,13,17,22,33,28)(55,56,57,58,59)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×C₁₃×D₄₁⋊C₅, C₅×C₁₃⋊(D₄₁⋊C₅), C₅×D₁₃×C₄₁⋊C₅, …
maximal normal subgroup of	C₅×C₁₃×D₄₁⋊C₅, C₅×C₁₃⋊(D₄₁⋊C₅), C₅×D₁₃×C₄₁⋊C₅, …
maximal quotient of	C₂×C₅×C₁₃×C₄₁⋊C₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13325, 4);

C5×C13×C41⋊C5 #27211019