C₃×#89758 #13055415

order	1344
isoclinism	isoclinic class of #89428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#89758

Splittings

Hall	C₃×C₇ ⋊ #400
Sylow	C₇ ⋊ C₃×#400

Subgroups

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator, cosocle	C₂³×C₇	C₂³×C₃
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃×C₇	C₂³
hypercenter	C₂×C₃×C₄⋊C₄	D₇
hypocenter	C₇	C₃×#400
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊C₄

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

normal subgroups only

This group has 268 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) = 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)(12,14)(23,25,26,29)(24,27,28,30);(15,17,18,22)(16,19,20,21)(25,29)(27,30);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16,18,20)(17,21,22,19)(23,24,26,28)(25,27,29,30)

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, 4704);