C₇×#5923 #13062099

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

solvable

Aliases

direct product

C₇×#5923

Splittings

Fitting

C₂×C₇×#41 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂×S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂³×C₇
Frattini	C₂	C₂²×C₇×S₄
Fitting	C₂×C₇×#41	D₃
socle	C₂×C₇	C₂²×S₄
hypercenter	C₂×C₄×C₇	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇

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

normal subgroups only

This group has 288 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)(10,18,33)(13,23,24)(14,36,38,25,26,39)(17,29)(20,34,35)(27,37);(8,10,13,20)(9,14,15,25)(11,18,23,34)(12,19,21,32)(16,26,28,36)(17,27,29,37)(22,33,24,35)(30,38,31,39);(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)(33,39)(34,37)(35,38)

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11388);