C₃×#89488 #13055149

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

Aliases

direct product

C₃×#89488

Splittings

Hall	C₃×C₇ ⋊ #441
Sylow	C₇ ⋊ C₃×#441

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×D₂₈
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×#97	C₂
socle, prime radical	C₂²×C₃×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₃×#97	D₇
hypocenter	C₇	C₃×#441
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#97

216 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 30
transitive degree	μ_t(G) = 1344

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3)(4,5,7,9,10,8,6)(11,12,13,14)(17,25)(20,29)(23,30)(26,28);(15,17,24,28,19,25,18,26)(16,20,27,30,22,29,21,23);(5,6)(7,8)(9,10)(15,16,19,22)(17,23,25,30)(18,27,24,21)(20,28,29,26)

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, 4438);