C₃×#89807 #13055468

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

Aliases

direct product

C₃×#89807

Splittings

Hall	C₃×C₇ ⋊ #482
Sylow	C₇ ⋊ C₃×#482

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting, prime radical	C₃×C₇×#121	C₂
socle	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₃×#121	D₇
hypocenter	C₇	C₃×#482
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#121

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

normal subgroups only

This group has 212 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)(13,17)(15,18)(20,25)(21,22)(24,26);(11,13)(12,15)(14,17)(16,18)(19,21,23,22)(20,24,26,25);(5,6)(7,8)(9,10)(11,12,14,16)(13,15,17,18)(19,20,23,26)(21,24,22,25)

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, 4757);