C₅×C₃⋊D₃×C₅₃ #18012018

order	4770
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₅₃ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₅×C₃⋊D₃×C₅₃
unambiguous semidirect product	C₃⋊(C₅×D₃×C₅₃)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₅₃ ⋊ C₂
derived, Sylow	C₃² ⋊ C₂×C₅×C₅₃
Hall	C₃²×C₅₃ ⋊ C₂×C₅
Hall	C₃²×C₅ ⋊ C₂×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₅₃	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂×C₅×C₅₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₅₃	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₅×C₅₃

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 2385

A minimal faithful permutation representation — 3 generators on 64 points:

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

Conjugacy classes & character table

1590 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃⋊D₃×C₂₅×C₅₃, C₅²×C₃⋊D₃×C₅₃, C₅×C₅₃×C₃⋊D₁₅, …
maximal normal subgroup of	C₃⋊D₃×C₂₅×C₅₃, C₅²×C₃⋊D₃×C₅₃, C₅×C₇×C₃⋊D₃×C₅₃
maximal quotient of	C₃⋊D₃×C₂₅×C₅₃, C₅²×C₃⋊D₃×C₅₃, C₅×C₅₃×C₃⋊D₁₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4770, 17);

C5×C3⋊D3×C53 #18012018