C₅₃×C₃⋊D₁₅ #18012020

order	4770
isoclinism	isoclinic class of C₃⋊D₁₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₅₃ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₅₃×C₃⋊D₁₅
unambiguous semidirect product	C₃⋊(D₁₅×C₅₃)
unambiguous semidirect product	C₅⋊(C₃⋊D₃×C₅₃)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₅₃ ⋊ C₂
derived, Hall	C₃²×C₅ ⋊ C₂×C₅₃
Hall	C₃²×C₅₃ ⋊ D₅
Hall	C₅×C₅₃ ⋊ C₃⋊D₃
Sylow	C₃² ⋊ D₅×C₅₃
Sylow	C₅ ⋊ C₃⋊D₃×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	C₃⋊D₁₅
commutator, cosocle, hypocenter	C₃²×C₅	C₂×C₅₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₅₃	C₂
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ 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:

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

Conjugacy classes & character table

1272 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅₃×C₃⋊D₇₅, C₅×C₅₃×C₃⋊D₁₅, C₅₃×C₃⋊(C₃⋊(C₅⋊D₅)), …
maximal normal subgroup of	C₅×C₅₃×C₃⋊D₁₅, C₇×C₅₃×C₃⋊D₁₅
maximal quotient of	C₅₃×C₃⋊D₇₅, C₅×C₅₃×C₃⋊D₁₅, C₅₃×C₃⋊(C₃⋊(C₅⋊D₅)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4770, 19);

C53×C3⋊D15 #18012020