C₂×D₄×#737 #13048978

order	1296
nilpotency class	2
isoclinism	isoclinic class of D₄×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃³ · Schur multiplier (H₂) = C₂³×C₃²

Aliases

direct product	C₂×D₄×#737
unambiguous semidirect product	C₄⋊(C₂²×#737)

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂²×C₃²
commutator, Frattini, cosocle	C₂×C₃	C₂³×C₃³
socle	C₂²×C₃	C₂²×C₃³
prime radical	C₂×D₄×H(3,𝔽₃)	C₃
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ G

891 conjugacy classes of subgroups; only the 551 normal subgroups are shown.

normal subgroups only

This group has 891 conjugacy classes of subgroups — too many to draw. Only its 551 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) = 33
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 33 points:

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

Conjugacy classes & character table

330 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×D₄×#4207
maximal subgroup of	C₂×D₄×#4207
maximal normal subgroup of	C₂×D₄×#4207
maximal quotient of	C₂×D₄×C₉×3₋¹⁺²

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 2696);