C₃×#90618 #13062343

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₃×#90618

Splittings

Fitting	C₃×C₇×H(5,𝔽₂) ⋊ C₂
Hall	C₃×C₇ ⋊ #585
Sylow	C₇ ⋊ C₃×#585

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂⁴×D₇
commutator, cosocle	C₂×C₇	C₂⁵×C₃
Frattini	C₂	C₂⁴×C₃×D₇
Fitting	C₃×C₇×H(5,𝔽₂)	C₂
socle	C₂×C₃×C₇	C₂⁵
hypercenter	C₃×H(5,𝔽₂)	D₇
hypocenter	C₇	C₃×#585
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×H(5,𝔽₂)

1620 conjugacy classes of subgroups; only the 886 normal subgroups are shown.

normal subgroups only

This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 886 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) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11632);