C₂×D₇₅₉ #15690109

order	3036
automorphism	Aut(G) order 667920
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₅₁₈
direct product	C₂×D₇₅₉
unambiguous semidirect product	(C₂×C₃)⋊D₂₅₃
unambiguous semidirect product	(C₂×C₁₁)⋊D₆₉
unambiguous semidirect product	(C₂×C₂₃)⋊D₃₃

Splittings

Fitting, socle	C₂×C₃×C₁₁×C₂₃ ⋊ C₂
derived, Hall	C₃×C₁₁×C₂₃ ⋊ C₂²
Hall	C₁₁×C₂₃ ⋊ C₂×D₃
Hall	C₃×C₂₃ ⋊ C₂×D₁₁
Hall	C₃×C₁₁ ⋊ C₂×D₂₃
Sylow	C₂₃ ⋊ C₂×D₃₃
Sylow	C₁₁ ⋊ C₂×D₆₉
Sylow	C₃ ⋊ C₂×D₂₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₇₅₉
commutator, cosocle, hypocenter	C₃×C₁₁×C₂₃	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₁×C₂₃	C₂
LCS	G ▷ C₃×C₁₁×C₂₃
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 1518

A minimal faithful permutation representation — 2 generators on 39 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);(4,5)(7,16)(8,15)(9,14)(10,13)(11,12)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₃⋊(C₂₃⋊AGL(1,𝔽₁₁)), D₅×D₇₅₉, C₂×D₃₇₉₅, …
maximal normal subgroup of	C₂×C₃⋊(C₂₃⋊AGL(1,𝔽₁₁)), C₂×C₅×D₇₅₉, C₂×C₇×D₇₅₉, …
maximal quotient of	D₅×D₇₅₉, C₂×D₃₇₉₅, C₂×C₅×D₇₅₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3036, 28);

C2×D759 #15690109