C₂²⋊C₄×#737 #13048036

order	1296
nilpotency class	2
isoclinism	isoclinic class of D₄×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³×C₄ · Schur multiplier (H₂) = C₂²×C₃²

Aliases

direct product

C₂²⋊C₄×#737

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂²×C₃²
commutator	C₂×C₃	C₂×C₃³×C₄
Frattini, socle, cosocle	C₂²×C₃	C₂²×C₃³
prime radical	C₂³×H(3,𝔽₃)	C₂×C₃
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ G

561 conjugacy classes of subgroups; only the 319 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 319 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) = 35
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

330 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 1754);