C₃×#89561 #13055251

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

Aliases

direct product

C₃×#89561

Splittings

Hall	C₃×C₇ ⋊ #441
Sylow	C₇ ⋊ C₃×#441

Subgroups

	subgroup	G/H
center	C₂²×C₃	D₄×D₇
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×#103	C₂
socle, prime radical	C₂²×C₃×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₃×#103	D₇
hypocenter	C₇	C₃×#441
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#103

216 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 98 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)(15,17,19,25)(16,20,22,23)(18,27,26,24)(21,30,29,28);(11,13,14,12)(17,24)(18,26)(20,28)(21,29)(23,30)(25,27);(5,6)(7,8)(9,10)(11,12,14,13)(15,16,19,22)(17,23,25,20)(18,21,26,29)(24,30,27,28)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4540);