C₃×#89544 #13055204

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

Aliases

direct product

C₃×#89544

Splittings

Fitting	C₃×C₇×#102 ⋊ C₂
Hall	C₃×C₇ ⋊ #416
Sylow	C₇ ⋊ C₃×#416

Subgroups

	subgroup	G/H
center	C₂²×C₃	D₄×D₇
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×#102	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₃×#1159	C₂
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₃×#102	D₇
hypocenter	C₇	C₃×#416
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#102

316 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 110 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)(15,17,18,26,20,27,19,25)(16,22,23,29,21,28,24,30);(16,21)(17,25)(18,19)(22,29)(26,27)(28,30);(5,6)(7,8)(9,10)(15,16)(17,22)(18,23)(19,24)(20,21)(25,30)(26,29)(27,28)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4493);