C₃×#90510 #13061161

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#90510

Splittings

Hall	C₃×C₇ ⋊ #568
Sylow	C₇ ⋊ C₃×#568

Subgroups

	subgroup	G/H
center	C₃×C₄	C₂³×D₇
commutator	C₂×C₇	C₂³×C₃×C₄
Frattini	C₄	C₂³×C₃×D₇
Fitting, prime radical	C₂×C₃×C₇×#41	C₂
socle	C₂×C₃×C₇	C₂³×C₄
cosocle	C₄×C₇	C₂⁴×C₃
hypercenter	C₂×C₃×#41	D₇
hypocenter	C₇	C₃×#568
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×#41

516 conjugacy classes of subgroups; only the 374 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 374 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) = 336

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

282 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10450);