C₃×#88251 #11929276

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

Aliases

direct product

C₃×#88251

Splittings

Fitting	C₂×C₃×D₄×C₁₃ ⋊ 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	C₂×C₃×D₄×C₁₃	C₂
socle	C₂×C₃×C₁₃	C₂⁴
hypercenter	C₂×C₃×D₄	D₁₃
hypocenter	C₁₃	C₃×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×D₄

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

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, 1388);