C₃×#90260 #13060911

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#90260

Splittings

Hall	C₃×C₇ ⋊ #557
Sylow	C₇ ⋊ C₃×#557

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₄×C₇×Q₈	C₂
socle, prime radical	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×C₄×Q₈	D₇
hypocenter	C₇	C₃×#557
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈

400 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 226 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 — 4 generators on 26 points:

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

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10200);