C₄×C₅²×C₄₃⋊C₃ #27166386

order	12900
isoclinism	isoclinic class of C₄₃⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₅² · Schur multiplier (H₂) = C₅

Aliases

direct product	C₄×C₅²×C₄₃⋊C₃
unambiguous semidirect product	(C₄×C₄₃)⋊(C₃×C₅²)
unambiguous semidirect product	(C₄×C₅×C₄₃)⋊(C₃×C₅)

Splittings

Fitting, Hall	C₄×C₅²×C₄₃ ⋊ C₃
derived, Sylow	C₄₃ ⋊ C₃×C₄×C₅²
Hall	C₅²×C₄₃ ⋊ C₃×C₄
Hall	C₄×C₄₃ ⋊ C₃×C₅²

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₅²	C₄₃⋊C₃
commutator, hypocenter	C₄₃	C₃×C₄×C₅²
Frattini	C₂	C₂×C₅²×C₄₃⋊C₃
Fitting	C₄×C₅²×C₄₃	C₃
socle	C₂×C₅²×C₄₃	C₂×C₃
prime radical	C₂×C₅²×C₄₃⋊C₃	C₂
cosocle	C₂×C₄₃	C₂×C₃×C₅²
LCS	G ▷ C₄₃
UCS	C₁ ◁ C₄×C₅²

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 4300

A minimal faithful permutation representation — 2 generators on 57 points:

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

Conjugacy classes & character table

1700 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 55);

C4×C52×C43⋊C3 #27166386