C₇×#4909 #13055944

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#4909

Splittings

Fitting	C₃×C₇×M₃₂ ⋊ C₂
Hall	C₃×C₇ ⋊ #510
Sylow	C₃ ⋊ C₇×#510

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂×D₂₄
commutator, cosocle	C₃×C₈	C₂³×C₇
Frattini	C₈	C₂²×D₃×C₇
Fitting	C₃×C₇×M₃₂	C₂
socle	C₂×C₃×C₇	C₂×D₈
prime radical	C₇×#904	C₂
hypercenter	C₇×M₃₂	D₃
hypocenter	C₃	C₇×#510
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₈ ◁ C₇×M₃₂

164 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 42
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2,3,4,5,6,7)(8,9,10)(11,14,23,38,24,26,15,27,16,28,35,41,13,22,30,42)(12,18,33,39,17,31,19,25,21,34,29,37,20,32,36,40);(11,13,16,24)(12,17,21,20)(14,26,28,22)(15,23,30,35)(18,32,34,31)(19,29,36,33)(25,39,40,37)(27,41,42,38);(9,10)(11,12)(13,17)(14,25)(15,29)(16,21)(18,27)(19,35)(20,24)(22,37)(23,36)(26,39)(28,40)(30,33)(31,38)(32,41)(34,42)

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5233);