C₇×#5406 #13057126

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product

C₇×#5406

Splittings

Fitting

C₇×#131 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₄⋊S₄
commutator, cosocle	#41⋊C₃	C₂²×C₇
Frattini	C₄	C₂×C₇×S₄
Fitting	C₇×#131	D₃
prime radical	C₇×#986	C₂
hypercenter	C₇×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₇×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈

124 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 26 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)(8,15,34,14,23,31,18,38,22,9,19,37,13,30,35,27,11,29,10,25,24,17,32,39)(12,16,20,26,21,28,33,36);(9,14)(10,18)(11,12)(15,26)(16,25)(17,27)(19,33)(20,32)(21,23)(22,24)(28,38)(29,39)(30,36)(31,37)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6415);