C₃×#89489 #13055150

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

Aliases

direct product

C₃×#89489

Splittings

Fitting	C₃×C₇×#97 ⋊ C₂
Hall	C₃×C₇ ⋊ #444
Sylow	C₇ ⋊ C₃×#444

Subgroups

	subgroup	G/H
center	C₃×C₄	C₂×D₂₈
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×#97	C₂
socle	C₂×C₃×C₇	C₄×D₄
prime radical	C₃×#6467	C₂
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₃×#97	D₇
hypocenter	C₇	C₃×#444
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×#97

208 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 26 points:

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

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, 4439);