C₇×#5404 #13057124

order	1344
isoclinism	isoclinic class of #226
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product

C₇×#5404

Splittings

Fitting

C₇×#131 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₇×C₈	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂×C₄×C₇
Frattini	C₄	C₂×C₇×S₄
Fitting	C₇×#131	D₃
socle	C₂×C₇	C₄×S₄
prime radical	C₇×#985	C₂
cosocle	#41⋊C₃	C₂²×C₇
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₇×C₈

128 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 30 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 — 2 generators on 39 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6413);