C₅×C₃⋊(C₄₃⋊AGL(1,𝔽₅)) #27166454

order	12900
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₃⋊(C₄₃⋊AGL(1,𝔽₅))
unambiguous semidirect product	C₃⋊(C₅×C₄₃⋊AGL(1,𝔽₅))
unambiguous semidirect product	C₄₃⋊(C₅×ΓL(1,𝔽₁₆))

Splittings

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊(C₄₃⋊AGL(1,𝔽₅))
commutator, hypocenter	C₃×C₅×C₄₃	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₄₃	C₄
prime radical	C₃×C₅×D₅×C₄₃	C₂
cosocle	C₃×D₅×C₄₃	C₂×C₅
LCS	G ▷ C₃×C₅×C₄₃
UCS	C₁ ◁ C₅

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 2580

A minimal faithful permutation representation — 2 generators on 56 points:

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

Conjugacy classes & character table

1305 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 123);

Fitting, socle, Hall	C₃×C₅²×C₄₃ ⋊ C₄
derived	C₃×C₅×C₄₃ ⋊ C₄×C₅
Hall	C₅²×C₄₃ ⋊ Dic₃
Hall	C₃×C₄₃ ⋊ C₅×AGL(1,𝔽₅)
Hall	C₃×C₅² ⋊ Dic₄₃
Sylow	C₄₃ ⋊ C₅×ΓL(1,𝔽₁₆)
Sylow	C₅² ⋊ Dic₁₂₉
Sylow	C₃ ⋊ C₅×C₄₃⋊AGL(1,𝔽₅)

C5×C3⋊(C43⋊AGL(1,𝔽5)) #27166454