C₇₃⋊C₃ #6344

order	219
isoclinism	isoclinic class of C₇₃⋊C₃
automorphism	Aut(G) = AGL(1,𝔽₇₃) (order 5256) · Inn(G) = G · Out(G) = C₃×C₈
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₇₃⋊C₃

Splittings

Fitting, socle, derived, Sylow

C₇₃ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇₃	C₃
LCS	G ▷ C₇₃
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 73

A minimal faithful permutation representation — 2 generators on 73 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,50)(28,40,47)(31,38,44)(34,45,42)(36,48,39)(41,53,62)(43,54,59)(46,58,57)(49,60,55)(51,63,52)(56,66,69)(61,68,67)(64,70,65)(71,73,72);(1,2,4,8,15,25,38,17,3,6,11,18,28,41,54,27,7,13,21,31,43,56,66,40,14,23,34,46,50,62,69,53,24,36,49,61,63,35,47,59,37,51,64,71,70,48,22,32,44,57,67,72,73,60,33,12,19,29,42,55,65,68,45,20,5,9,16,26,39,52,58,30,10)

Polycyclic presentation:

3,73|b^a=b^8

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇₃⋊C₃, C₂×C₇₃⋊C₃, #10620323, …
maximal normal subgroup of	D₇₃⋊C₃, C₂×C₇₃⋊C₃, #10620323, …
maximal quotient of	C₂×C₇₃⋊C₃, #10620321, C₃×C₇₃⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(219, 1);

C73⋊C3 #6344