C₃×#89897 #13055558

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

Aliases

direct product

C₃×#89897

Splittings

Hall	C₃×C₇ ⋊ #430
Sylow	C₇ ⋊ C₃×#430

Subgroups

	subgroup	G/H
center	C₃×C₄	C₂×D₂₈
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 134 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 134 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)(11,13,15,23,16,21,14,22)(12,17,18,20,19,25,24,26);(13,21)(17,25)(20,26)(22,23);(5,6)(7,8)(9,10)(11,12,14,18,16,19,15,24)(13,20,22,25,21,26,23,17)

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, 4847);