C₃×#89533 #13055192

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#89533

Splittings

Hall	C₃×C₇ ⋊ #476
Sylow	C₇ ⋊ C₃×#476

Subgroups

	subgroup	G/H
center	C₂²×C₃	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×#102	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₃×C₇×D₄	C₂²
hypercenter	C₃×#102	D₇
hypocenter	C₇	C₃×#476
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#102

216 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 86 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)(11,13,14,18)(12,15,16,17)(20,23)(21,25)(24,26);(19,21)(20,24)(22,26)(23,25);(5,6)(7,8)(9,10)(11,12,14,16)(13,17,18,15)(19,20,22,23)(21,24,26,25)

Conjugacy classes & character table

183 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4481);