C₃×#90525 #13061176

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#90525

Splittings

Fitting	C₂×C₃×C₇×#41 ⋊ C₂
Hall	C₃×C₇ ⋊ #550
Sylow	C₇ ⋊ C₃×#550

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₂×C₃×C₇×#41	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂×C₃×#41	D₇
hypocenter	C₇	C₃×#550
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41

624 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 22
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 4 generators on 22 points:

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

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, 10465);