C₃×#89764 #13055411

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

Aliases

direct product

C₃×#89764

Splittings

Fitting	C₂×C₃×C₇×C₄⋊C₄ ⋊ C₂
Hall	C₃×C₇ ⋊ #399
Sylow	C₇ ⋊ C₃×#399

Subgroups

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator, cosocle	C₂³×C₇	C₂³×C₃
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×C₄⋊C₄	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×C₃×D₇	C₂²
hypercenter	C₂×C₃×C₄⋊C₄	D₇
hypocenter	C₇	C₃×#399
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊C₄

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

normal subgroups only

This group has 372 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)(16,18)(19,21,22,25)(20,24,23,26);(15,16,17,18)(20,23)(24,26);(5,6)(7,8)(9,10)(11,12)(13,14)(16,18)(19,20)(21,24)(22,23)(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, 4700);