C₃×#89867 #13055528

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

Aliases

direct product

C₃×#89867

Splittings

Fitting	C₃×C₇×C₄⋊Q₈ ⋊ C₂
Hall	C₃×C₇ ⋊ #456
Sylow	C₇ ⋊ C₃×#456

Subgroups

	subgroup	G/H
center	C₂×C₃	#6500
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×C₄⋊Q₈	C₂
socle	C₂×C₃×C₇	#120
prime radical	C₃×#6551	C₂
hypercenter	C₃×C₄⋊Q₈	D₇
hypocenter	C₇	C₃×#456
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×C₄⋊Q₈

284 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 336

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

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

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, 4817);