C₄×C₂₅×C₄₃⋊C₃ #27166336

order	12900
isoclinism	isoclinic class of C₄₃⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₂₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₄×C₂₅×C₄₃⋊C₃
unambiguous semidirect product	(C₄×C₄₃)⋊(C₃×C₂₅)
unambiguous semidirect product	(C₄×C₂₅×C₄₃)⋊C₃

Splittings

Fitting, Hall	C₄×C₂₅×C₄₃ ⋊ C₃
derived, Sylow	C₄₃ ⋊ C₃×C₄×C₂₅
Hall	C₂₅×C₄₃ ⋊ C₃×C₄
Hall	C₄×C₄₃ ⋊ C₃×C₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₂₅	C₄₃⋊C₃
commutator, hypocenter	C₄₃	C₃×C₄×C₂₅
Frattini	C₂×C₅	C₂×C₅×C₄₃⋊C₃
Fitting	C₄×C₂₅×C₄₃	C₃
socle, cosocle	C₂×C₅×C₄₃	C₂×C₃×C₅
prime radical	C₂×C₅×C₄₃⋊C₃	C₂×C₅
LCS	G ▷ C₄₃
UCS	C₁ ◁ C₄×C₂₅

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 4300

A minimal faithful permutation representation — 2 generators on 72 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)(44,52,65,55,66,46,57,47,60,68,49,62,51,64,54,53,45,56,67,59,58,48,61,50,63)(69,70,71,72);(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)

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, 5);

C4×C25×C43⋊C3 #27166336