C₃⋊C₃₂ #794

order	96
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×D₃×C₈ (order 96) · Inn(G) = D₃ · Out(G) = C₂×C₈
holomorph	G ⋊ Aut(G) = #72222660
homology	Abelianization (H₁) = C₃₂ · Schur multiplier (H₂) = C₁

Aliases

C₃⋊C₃₂

C₃ ⋊ C₃₂

	subgroup	G/H
center, Frattini, hypercenter	C₁₆	D₃
commutator, hypocenter	C₃	C₃₂
Fitting, cosocle	C₃×C₁₆	C₂
socle, prime radical	C₂×C₃	C₁₆
LCS	G ▷ C₃
UCS	C₁ ◁ C₁₆

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 35 points:

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

Polycyclic presentation:

2^5,3|a^2=b,b^2=c,c^2=d,d^2=e,f^a=f^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 1);