C₃×#89742 #13055395

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#89742

Splittings

Hall	C₃×C₇ ⋊ #484
Sylow	C₇ ⋊ C₃×#484

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×C₄⋊C₄	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂³×C₃×C₇	D₄
hypercenter	C₂×C₃×C₄⋊C₄	D₇
hypocenter	C₇	C₃×#484
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄⋊C₄

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

normal subgroups only

This group has 208 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)(11,13,14,17)(12,16,15,18)(20,24)(21,26)(23,25);(12,15)(16,18)(20,23)(24,25);(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20,22,23)(21,24,26,25)

Conjugacy classes & character table

237 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4684);