C₃×#90285 #13060936

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₃×#90285

Splittings

Fitting	C₃×C₇×#120 ⋊ C₂
Hall	C₃×C₇ ⋊ H(3,𝔽₄)
Sylow	C₇ ⋊ C₃×H(3,𝔽₄)

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₇×#120	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×#120	D₇
hypocenter	C₇	C₃×H(3,𝔽₄)
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#120

520 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 182 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 — 4 generators on 26 points:

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

Conjugacy classes & character table

183 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10225);