C₄₉×C₃₁⋊C₃ #17986944

order	4557
isoclinism	isoclinic class of C₃₁⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₄₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₄₉×C₃₁⋊C₃
unambiguous semidirect product	C₃₁⋊(C₃×C₄₉)

Splittings

Fitting, Hall	C₃₁×C₄₉ ⋊ C₃
derived, Sylow	C₃₁ ⋊ C₃×C₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₄₉	C₃₁⋊C₃
commutator, hypocenter	C₃₁	C₃×C₄₉
Frattini	C₇	C₇×C₃₁⋊C₃
Fitting	C₃₁×C₄₉	C₃
socle, cosocle	C₇×C₃₁	C₃×C₇
prime radical	C₇×C₃₁⋊C₃	C₇
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₄₉

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 1519

A minimal faithful permutation representation — 2 generators on 80 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49)(51,52,54)(53,56,61)(55,59,58)(57,63,70)(60,66,68)(62,64,65)(67,73,77)(69,74,75)(71,78,72)(76,79,80);(50,51,53,57,64,54,58,65,72,78,61,68,75,63,71,70,77,80,73,56,62,69,76,79,66,52,55,60,67,74,59)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁⋊C₃×C₄₉⋊C₃, C₃×C₄₉×C₃₁⋊C₃, C₄₉×C₂²⋊(C₃₁⋊C₃), …
maximal normal subgroup of	C₃₁⋊C₃×C₄₉⋊C₃, C₃×C₄₉×C₃₁⋊C₃, C₅×C₄₉×C₃₁⋊C₃, …
maximal quotient of	C₄₉×C₃₁⋊C₉, C₃×C₄₉×C₃₁⋊C₃, C₄₉×C₂²⋊(C₃₁⋊C₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4557, 3);

C49×C31⋊C3 #17986944