C₃×#89896 #13055557

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

Aliases

direct product

C₃×#89896

Splittings

Hall	C₃×C₇ ⋊ #430
Sylow	C₇ ⋊ C₃×#430

Subgroups

	subgroup	G/H
center	C₃×C₄	C₂×C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×M₁₆	C₂
socle	C₂×C₃×C₇	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₇	Q₈
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₂×C₃×M₁₆	D₇
hypocenter	C₇	C₃×#430
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×M₁₆

204 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 118 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)(11,13,14,22,16,23,15,24)(12,18,19,21,17,25,20,26);(12,17)(18,25)(19,20)(21,26);(5,6)(7,8)(9,10)(11,12,15,20,16,17,14,19)(13,21,22,25,23,26,24,18)

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, 4846);