C₇₉⋊C₃ #6633

order	237
isoclinism	isoclinic class of C₇₉⋊C₃
automorphism	Aut(G) = AGL(1,𝔽₇₉) (order 6162) · Inn(G) = G · Out(G) = C₂×C₁₃
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₇₉⋊C₃

Splittings

Fitting, socle, derived, Sylow

C₇₉ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇₉	C₃
LCS	G ▷ C₇₉
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 79

A minimal faithful permutation representation — 2 generators on 79 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)(28,40,48)(31,43,45)(34,46,42)(36,38,39)(41,53,63)(44,56,61)(47,59,58)(49,54,55)(51,64,52)(57,65,72)(60,68,70)(62,66,67)(69,73,77)(71,74,75)(76,78,79);(1,2,4,8,15,25,38,22,32,45,58,67,53,24,36,49,62,71,76,78,68,43,17,3,6,11,18,28,41,54,33,12,19,29,42,55,37,51,50,63,72,77,79,73,56,27,7,13,21,31,44,57,66,46,20,5,9,16,26,39,52,64,35,48,61,70,75,65,40,14,23,34,47,60,69,74,59,30,10)

Polycyclic presentation:

3,79|b^a=b^23

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇₉⋊C₃, C₂×C₇₉⋊C₃, C₃×C₇₉⋊C₃, …
maximal normal subgroup of	D₇₉⋊C₃, C₂×C₇₉⋊C₃, C₃×C₇₉⋊C₃, …
maximal quotient of	C₂×C₇₉⋊C₃, C₇₉⋊C₉, C₃×C₇₉⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(237, 1);

C79⋊C3 #6633