C₃×#89776 #13055438

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#89776

Splittings

Hall	C₃×C₇ ⋊ #419
Sylow	C₇ ⋊ C₃×#419

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×#117	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂³×C₃×C₇	D₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₃×#117	D₇
hypocenter	C₇	C₃×#419
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#117

300 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 30
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3)(4,5,7,9,10,8,6)(11,12,13,14)(16,21)(17,24)(18,26)(20,27)(22,23)(28,29);(15,17)(16,20)(18,25)(19,24)(21,29)(22,27)(23,28)(26,30);(5,6)(7,8)(9,10)(15,16,19,23)(17,20,24,28)(18,21,26,22)(25,29,30,27)

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, 4727);