C₆₇⋊C₃ #6066

order	201
isoclinism	isoclinic class of C₆₇⋊C₃
automorphism	Aut(G) = AGL(1,𝔽₆₇) (order 4422) · Inn(G) = G · Out(G) = C₂×C₁₁
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₆₇⋊C₃

Splittings

Fitting, socle, derived, Sylow

C₆₇ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₆₇	C₃
LCS	G ▷ C₆₇
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 67

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

Polycyclic presentation:

3,67|b^a=b^37

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₆₇⋊C₃, C₂×C₆₇⋊C₃, C₃×C₆₇⋊C₃, …
maximal normal subgroup of	D₆₇⋊C₃, C₂×C₆₇⋊C₃, C₃×C₆₇⋊C₃, …
maximal quotient of	C₂×C₆₇⋊C₃, C₆₇⋊C₉, C₃×C₆₇⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(201, 1);

C67⋊C3 #6066