C₃×C₇×#433 #13056944

order	1344
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃×C₇×#433

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄×C₇	C₂³
commutator	C₂²	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂³×C₃×C₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₇ ◁ G

192 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 132 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)(11,12)(13,16)(14,15)(17,18)(19,20,22,25,23,26,29,32)(21,24,28,31,27,30,33,34);(12,15)(16,18)(19,21)(20,24)(22,28)(23,27)(25,31)(26,30)(29,33)(32,34);(11,13,14,17)(12,16,15,18)(21,27)(24,30)(28,33)(31,34)

Conjugacy classes & character table

588 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6233);