C₃×#89549 #13055208

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

Aliases

direct product

C₃×#89549

Splittings

Fitting	C₃×C₇×#102 ⋊ C₂
Hall	C₃×C₇ ⋊ #481
Sylow	C₇ ⋊ C₃×#481

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	C₃×#6501	C₂
hypercenter	C₃×#102	D₇
hypocenter	C₇	C₃×#481
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#102

228 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 78 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)(12,16)(15,18)(20,23)(22,26)(24,25);(11,13,14,17)(12,15,16,18)(19,20,21,24)(22,23,26,25);(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)

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, 4497);