C₃×C₇×#410 #13056921

order	1344
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃×C₇×#410

Subgroups

	subgroup	G/H
center	C₃×C₄×C₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂³	C₂³×C₃×C₇
socle	C₂×C₃×C₇	C₂×C₂²⋊C₄
prime radical	C₂×C₃×C₇×#41	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₇ ◁ C₂²×C₃×C₄×C₇ ◁ G

316 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 156 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)(11,12)(13,19,21,16)(14,17)(15,18)(20,25,26,23)(22,24);(11,13)(12,16)(14,20)(15,21)(17,23)(18,19)(22,26)(24,25);(11,14,15,22)(12,17,18,24)(13,20,21,26)(16,23,19,25)

Conjugacy classes & character table

462 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6210);