C₃×#89852 #13055512

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#89852

Splittings

Hall	C₃×C₇ ⋊ #501
Sylow	C₇ ⋊ C₃×#501

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×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₃×C₄⋊Q₈	D₇
hypocenter	C₇	C₃×#501
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊Q₈

180 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 86 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)(11,13,14,18)(12,15,16,17)(20,24)(21,26)(23,25);(11,12,14,16)(13,17,18,15)(20,23)(24,25);(5,6)(7,8)(9,10)(19,20,21,25,22,23,26,24)

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, 4801);