C₅×C₃⋊D₆₉ #14523403

order	2070
isoclinism	isoclinic class of C₃⋊D₆₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₅×C₃⋊D₆₉
unambiguous semidirect product	C₃⋊(C₅×D₆₉)
unambiguous semidirect product	C₂₃⋊(C₅×C₃⋊D₃)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₂₃ ⋊ C₂
derived, Hall	C₃²×C₂₃ ⋊ C₂×C₅
Hall	C₅×C₂₃ ⋊ C₃⋊D₃
Hall	C₃²×C₅ ⋊ D₂₃
Sylow	C₂₃ ⋊ C₅×C₃⋊D₃
Sylow	C₃² ⋊ C₅×D₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊D₆₉
commutator, cosocle, hypocenter	C₃²×C₂₃	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₂₃	C₂
LCS	G ▷ C₃²×C₂₃
UCS	C₁ ◁ C₅

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 1035

A minimal faithful permutation representation — 3 generators on 34 points:

(1,2,3,4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(30,31)(33,34);(6,7,9,11,13,15,17,19,21,23,25,27,28,26,24,22,20,18,16,14,12,10,8)(32,33,34);(29,30,31)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×C₃⋊D₃×D₂₃, D₅×C₃⋊D₆₉, D₁₁₅⋊(C₃⋊D₃), …
maximal normal subgroup of	C₅×C₃⋊D₃×D₂₃, D₅×C₃⋊D₆₉, D₁₁₅⋊(C₃⋊D₃), …
maximal quotient of	C₅×C₃⋊Dic₆₉, C₂×C₅×C₃⋊D₆₉, C₅×C₃⋊D₄₈₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2070, 18);

C5×C3⋊D69 #14523403