C₇×#4752 #13055787

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

Aliases

direct product

C₇×#4752

Splittings

Fitting	C₃×C₇×#101 ⋊ C₂
Hall	C₃×C₇ ⋊ #456
Sylow	C₃ ⋊ C₇×#456

Subgroups

	subgroup	G/H
center	C₂×C₇	#882
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#101	C₂
socle	C₂×C₃×C₇	#120
prime radical	C₇×#1007	C₂
hypercenter	C₇×#101	D₃
hypocenter	C₃	C₇×#456
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#101

284 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 3 generators on 26 points:

(1,2,3,4,5,6,7)(8,9,10)(11,14,15,24,16,25,26,17)(12,18,19,23,20,13,21,22);(11,13,16,18)(12,17,20,24)(14,19,25,21)(15,23,26,22);(9,10)(11,12)(13,17)(14,23)(15,21)(16,20)(18,24)(19,26)(22,25)

Conjugacy classes & character table

189 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5076);