C₇×C₃⋊D₆₉ #15668491

order	2898
isoclinism	isoclinic class of C₃⋊D₆₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₇×C₃⋊D₆₉
unambiguous semidirect product	C₃⋊(C₇×D₆₉)
unambiguous semidirect product	C₂₃⋊(C₇×C₃⋊D₃)

Splittings

Fitting, socle, Hall	C₃²×C₇×C₂₃ ⋊ C₂
derived, Hall	C₃²×C₂₃ ⋊ C₂×C₇
Hall	C₇×C₂₃ ⋊ C₃⋊D₃
Hall	C₃²×C₇ ⋊ D₂₃
Sylow	C₂₃ ⋊ C₇×C₃⋊D₃
Sylow	C₃² ⋊ C₇×D₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₇	C₃⋊D₆₉
commutator, cosocle, hypocenter	C₃²×C₂₃	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₂₃	C₂
LCS	G ▷ C₃²×C₂₃
UCS	C₁ ◁ C₇

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 1449

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,3,4,5,6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(32,33)(35,36);(8,9,11,13,15,17,19,21,23,25,27,29,30,28,26,24,22,20,18,16,14,12,10)(34,35,36);(31,32,33)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×C₃⋊D₃₄₅, C₅×C₇×C₃⋊D₆₉, C₄₉×C₃⋊D₆₉, …
maximal normal subgroup of	C₅×C₇×C₃⋊D₆₉, C₄₉×C₃⋊D₆₉, C₇²×C₃⋊D₆₉, …
maximal quotient of	C₇×C₃⋊D₃₄₅, C₅×C₇×C₃⋊D₆₉, C₄₉×C₃⋊D₆₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2898, 30);

C7×C3⋊D69 #15668491