C₃×#89746 #13055400

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

Aliases

direct product

C₃×#89746

Splittings

Hall	C₃×C₇ ⋊ #391
Sylow	C₇ ⋊ C₃×#391

Subgroups

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₃×C₄
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃×C₇	C₂³
cosocle	C₂³×C₇	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊C₄	D₇
hypocenter	C₇	C₃×#391
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊C₄

372 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 1344

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

264 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4689);