C₃⋊C₆₄ #4442

order	192
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×D₃×C₁₆ (order 192) · Inn(G) = D₃ · Out(G) = C₂×C₁₆
homology	Abelianization (H₁) = C₆₄ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₃⋊C₆₄

Splittings

derived, Sylow

C₃ ⋊ C₆₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃₂	D₃
commutator, hypocenter	C₃	C₆₄
Fitting, cosocle	C₃×C₃₂	C₂
socle, prime radical	C₂×C₃	C₃₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₃₂

14 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 2 generators on 67 points:

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

Polycyclic presentation:

2^6,3|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,g^a=g^2

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₆₄, #67432, C₂×C₃⋊C₆₄, …
maximal normal subgroup of	D₃×C₆₄, #67432, C₂×C₃⋊C₆₄, …
maximal quotient of	C₃⋊C₁₂₈, C₂×C₃⋊C₆₄, C₉⋊C₆₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1);

C3⋊C64 #4442