C₁₃×2₋¹⁺⁴⋊C₃ #11929258

order	1248
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₁₃×2₋¹⁺⁴⋊C₃
unambiguous semidirect product	2₋¹⁺⁴⋊(C₃×C₁₃)

Fitting, Hall	C₁₃×2₋¹⁺⁴ ⋊ C₃
Sylow	2₋¹⁺⁴ ⋊ C₃×C₁₃

	subgroup	G/H
center, socle	C₂×C₁₃	C₂²×A₄
commutator, cosocle, hypocenter	Q₈	C₂²×C₃×C₁₃
Frattini	C₂	C₂²×A₄×C₁₃
Fitting	C₁₃×2₋¹⁺⁴	C₃
hypercenter	D₄×C₁₃	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₁₃ ◁ D₄×C₁₃

92 conjugacy classes of subgroups.

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 208

A minimal faithful permutation representation — 2 generators on 29 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 1370);