C₇×#5405 #13057125

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

solvable

Aliases

direct product

C₇×#5405

Splittings

Fitting

C₇×#131 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂×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
hypercenter	C₇×C₈	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₇ ◁ C₇×C₈

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

normal subgroups only

This group has 124 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)(8,16,33,15,23,31,20,36,22,9,18,38,13,26,35,25,11,30,10,28,24,14,34,39)(12,17,19,29,21,27,32,37);(9,14)(11,12)(15,25)(16,27)(17,26)(18,19)(21,23)(22,24)(28,37)(29,36)(32,34)(33,35)

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, 6414);