C₁₇×D₁₁ #67324

order	374
isoclinism	isoclinic class of D₁₁
automorphism	Aut(G) = C₁₆×AGL(1,𝔽₁₁) (order 1760) · Inn(G) = D₁₁ · Out(G) = C₅×C₁₆
homology	Abelianization (H₁) = C₂×C₁₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₇×D₁₁
unambiguous semidirect product	C₁₁⋊(C₂×C₁₇)

Fitting, socle, Hall	C₁₁×C₁₇ ⋊ C₂
derived, Sylow	C₁₁ ⋊ C₂×C₁₇

	subgroup	G/H
center, hypercenter	C₁₇	D₁₁
commutator, cosocle, hypocenter	C₁₁	C₂×C₁₇
Frattini	C₁	G
Fitting, socle	C₁₁×C₁₇	C₂
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₁₇

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 187

A minimal faithful permutation representation — 2 generators on 28 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(374, 1);