C₂²×D₁₇ #3635

order	136
isoclinism	isoclinic class of D₁₇
automorphism	Aut(G) = S₄×AGL(1,𝔽₁₇) (order 6528) · Inn(G) = D₁₇ · Out(G) = C₈×S₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂²×D₁₇
unambiguous semidirect product	C₁₇⋊C₂³
unambiguous semidirect product	(C₂×C₁₇)⋊C₂²

Fitting, socle	C₂²×C₁₇ ⋊ C₂
derived, Sylow	C₁₇ ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂²	D₁₇
commutator, cosocle, hypocenter	C₁₇	C₂³
Frattini	C₁	G
Fitting, socle	C₂²×C₁₇	C₂
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂²

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 68

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2)(3,4)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21);(3,4)(5,6,8,10,12,14,16,18,20,21,19,17,15,13,11,9,7);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(136, 14);