C₇×Q₈⋊SL(2,𝔽₃) #13060799

order	1344
automorphism	?
homology	Abelianization (H₁) = C₃×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₇×Q₈⋊SL(2,𝔽₃)
unambiguous semidirect product	Q₈⋊(C₇×SL(2,𝔽₃))

Fitting, Hall	C₇×Q₈² ⋊ C₃
derived, Sylow	Q₈² ⋊ C₃×C₇

	subgroup	G/H
center, socle, hypercenter	C₂²×C₇	C₂²⋊A₄
commutator, cosocle, hypocenter	Q₈²	C₃×C₇
Frattini	C₂²	C₇×C₂²⋊A₄
Fitting	C₇×Q₈²	C₃
LCS	G ▷ Q₈²
UCS	C₁ ◁ C₂²×C₇

124 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10088);