C₂×D₄×C₂₇⋊C₃ #13047496

order	1296
nilpotency class	2
isoclinism	isoclinic class of D₄×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂×D₄×C₂₇⋊C₃
unambiguous semidirect product	C₄⋊(C₂²×C₂₇⋊C₃)
unambiguous semidirect product	(C₂×D₄×C₂₇)⋊C₃

Subgroups

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

324 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 209 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)(7,8,11,9,13,19,15,22,27,10,14,18,16,23,26,24,30,32,17,12,20,25,21,28,31,29,33);(3,4,5,6)(8,12,14)(11,18,20)(13,21,23)(19,26,28)(22,29,30)(27,32,33);(4,6)

Conjugacy classes & character table

330 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×D₄×D₂₇⋊C₃, #72276045, #72276234
maximal subgroup of	C₂×D₄×D₂₇⋊C₃, #127×C₂₇⋊C₃, #72276045, …
maximal normal subgroup of	C₂×D₄×D₂₇⋊C₃, #127×C₂₇⋊C₃, #72276045, …
maximal quotient of	#127×C₂₇⋊C₃, #120×C₂₇⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 1214);