C₇×#5423 #13057143

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product

C₇×#5423

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂×S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂×C₄×C₇
Frattini	C₄	C₂×C₇×S₄
Fitting	C₂×C₇×#41	D₃
socle	C₂×C₇	C₂×SL(2,ℤ/4)
prime radical	C₂×C₇×#41⋊C₃	C₂
cosocle	#41⋊C₃	C₂²×C₇
hypercenter	C₂×C₄×C₇	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇

144 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 38 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)(8,11,22)(9,26,30,14,16,31)(10,18,33)(13,23,24)(15,36,38,25,28,39)(17,27)(20,34,35)(29,37);(8,9,10,15,13,14,20,25)(11,17,18,29,23,27,34,37)(12,16,19,28,21,26,32,36)(22,31,33,39,24,30,35,38)

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, 6432);