C₇×#5926 #13062102

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

solvable

Aliases

direct product

C₇×#5926

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

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

normal subgroups only

This group has 304 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) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

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

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, 11391);