C₃×#89824 #13055480

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#89824

Splittings

Hall	C₃×C₇ ⋊ #470
Sylow	C₇ ⋊ C₃×#470

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂³×C₃×C₇	D₄
hypercenter	C₃×C₂²⋊Q₈	D₇
hypocenter	C₇	C₃×#470
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×C₂²⋊Q₈

220 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 672

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

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4769);