C₅₃×C₅²⋊C₃ #16860554

order	3975
isoclinism	isoclinic class of C₅²⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₅₃ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅₃×C₅²⋊C₃
unambiguous semidirect product	C₅²⋊(C₃×C₅₃)
unambiguous semidirect product	(C₅²×C₅₃)⋊C₃

Splittings

Fitting, socle, Hall	C₅²×C₅₃ ⋊ C₃
derived, Sylow	C₅² ⋊ C₃×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	C₅²⋊C₃
commutator, cosocle, hypocenter	C₅²	C₃×C₅₃
Frattini	C₁	G
Fitting, socle	C₅²×C₅₃	C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₅₃

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 795

A minimal faithful permutation representation — 2 generators on 68 points:

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

Conjugacy classes & character table

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

Relationships

derived subgroup of	#30629704, #3874⋊D₅₃, ((C₅⋊D₅)⋊C₃)⋊D₅₃, …
maximal subgroup of	C₃×C₅₃×C₅²⋊C₃, C₅₃×C₂²⋊(C₅²⋊C₃), C₇×C₅₃×C₅²⋊C₃, …
maximal normal subgroup of	C₃×C₅₃×C₅²⋊C₃, C₇×C₅₃×C₅²⋊C₃
maximal quotient of	C₅₃×C₅²⋊C₉, C₃×C₅₃×C₅²⋊C₃, C₅₃×C₂²⋊(C₅²⋊C₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3975, 3);

C53×C52⋊C3 #16860554