C₄₉×C₅²⋊C₃ #16814716

order	3675
isoclinism	isoclinic class of C₅²⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₄₉ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₄₉×C₅²⋊C₃
unambiguous semidirect product	C₅²⋊(C₃×C₄₉)

Splittings

Fitting, Hall	C₅²×C₄₉ ⋊ C₃
derived, Sylow	C₅² ⋊ C₃×C₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₄₉	C₅²⋊C₃
commutator, hypocenter	C₅²	C₃×C₄₉
Frattini	C₇	C₇×C₅²⋊C₃
Fitting	C₅²×C₄₉	C₃
socle, cosocle	C₅²×C₇	C₃×C₇
prime radical	C₇×C₅²⋊C₃	C₇
LCS	G ▷ C₅²
UCS	C₁ ◁ C₄₉

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 735

A minimal faithful permutation representation — 2 generators on 64 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)(50,51,53)(52,56,58)(54,59,62)(55,61,64)(57,60,63);(50,52,57,62,64)(51,56,60,54,55)(53,59,58,61,63)

Conjugacy classes & character table

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

Relationships

derived subgroup of	(C₄₉⋊C₃)⋊#3874, C₄₉⋊C₃×#3874, C₃×C₄₉⋊#3874
maximal subgroup of	C₅²⋊C₃×C₄₉⋊C₃, C₃×C₄₉×C₅²⋊C₃, C₄₉×C₂²⋊(C₅²⋊C₃)
maximal normal subgroup of	C₅²⋊C₃×C₄₉⋊C₃, C₃×C₄₉×C₅²⋊C₃
maximal quotient of	C₄₉×C₅²⋊C₉, C₃×C₄₉×C₅²⋊C₃, C₄₉×C₂²⋊(C₅²⋊C₃)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3675, 5);

C49×C52⋊C3 #16814716