C₇×#4814 #13055846

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#4814

Splittings

Fitting	C₃×C₇×#103 ⋊ C₂
Hall	C₃×C₇ ⋊ #485
Sylow	C₃ ⋊ C₇×#485

Subgroups

	subgroup	G/H
center	C₂²×C₇	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#103	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂²×D₃×C₇	D₄
hypercenter	C₇×#103	D₃
hypocenter	C₃	C₇×#485
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#103

192 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 34
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5135);