C₇×#5927 #13062103

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product

C₇×#5927

Splittings

Fitting

C₇×2₋¹⁺⁴ ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂²×S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂³×C₇
Frattini	C₂	C₂²×C₇×S₄
Fitting	C₇×2₋¹⁺⁴	D₃
hypercenter	C₇×D₄	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₇ ◁ C₇×D₄

280 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7)(8,11,22)(9,28,30,15,16,31)(10,32,34,18,19,35)(13,23,24)(14,25,37)(17,29)(20,33)(27,38,39);(8,10,13,18)(9,14,15,27)(11,19,23,32)(12,20,21,33)(16,25,28,38)(17,26,29,36)(22,34,24,35)(30,37,31,39);(8,9)(10,14)(11,17)(12,16)(13,15)(18,27)(19,26)(20,25)(21,28)(22,31)(23,29)(24,30)(32,36)(33,38)(34,39)(35,37)

Conjugacy classes & character table

161 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11392);