D₅₁×C₄₃⋊C₃ #27190486

order	13158
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅₁×C₄₃⋊C₃
unambiguous semidirect product	C₃⋊(D₁₇×C₄₃⋊C₃)
unambiguous semidirect product	C₁₇⋊(D₃×C₄₃⋊C₃)
unambiguous semidirect product	(C₄₃×D₅₁)⋊C₃

Splittings

Fitting, socle, derived	C₃×C₁₇×C₄₃ ⋊ C₂×C₃
Hall	C₃×C₁₇×C₄₃⋊C₃ ⋊ C₂
Hall	C₁₇×C₄₃ ⋊ C₃×D₃
Hall	C₃×C₄₃⋊C₃ ⋊ D₁₇
Sylow	C₄₃ ⋊ C₃×D₅₁
Sylow	C₁₇ ⋊ D₃×C₄₃⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₇×C₄₃	C₂×C₃
LCS	G ▷ C₃×C₁₇×C₄₃
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 2193

A minimal faithful permutation representation — 2 generators on 63 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₅₁×D₄₃⋊C₃, D₅₁⋊(D₄₃⋊C₃), D₃×D₁₇×C₄₃⋊C₃, …
maximal normal subgroup of	D₅₁×D₄₃⋊C₃, D₅₁⋊(D₄₃⋊C₃), D₃×D₁₇×C₄₃⋊C₃, …
maximal quotient of	C₄₃⋊C₃×Dic₅₁, C₂×D₅₁×C₄₃⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13158, 20);

D51×C43⋊C3 #27190486