C₃×#89445 #13055087

order	1344
isoclinism	isoclinic class of #89428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#89445

Splittings

Fitting	C₃×C₇×#95 ⋊ C₂
Hall	C₃×C₇ ⋊ #393
Sylow	C₇ ⋊ C₃×#393

Subgroups

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator, cosocle	C₂³×C₇	C₂³×C₃
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₃×C₇×#95	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×C₃×D₇	C₂²
hypercenter	C₃×#95	D₇
hypocenter	C₇	C₃×#393
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#95

404 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 672

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4376);