C₃×#88256 #11929281

order	1248
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#88256

Splittings

Fitting	C₃×C₁₃×#41 ⋊ C₂
Hall	C₃×C₁₃ ⋊ H(5,𝔽₂)
Sylow	C₁₃ ⋊ C₃×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂³×D₁₃
commutator, cosocle	C₂×C₁₃	C₂⁴×C₃
Frattini	C₂	C₂³×C₃×D₁₃
Fitting	C₃×C₁₃×#41	C₂
socle	C₂×C₃×C₁₃	C₂⁴
hypercenter	C₃×#41	D₁₃
hypocenter	C₁₃	C₃×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃×#41

332 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 312

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 1393);