C₂×C₄×#6895 #14517780

order	1944
nilpotency class	2
isoclinism	isoclinic class of C₃²⋊H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³×C₄ · Schur multiplier (H₂) = C₂×C₃²

Aliases

direct product

C₂×C₄×#6895

Subgroups

	subgroup	G/H
center	C₂×C₃²×C₄	C₃³
commutator	C₃²	C₂×C₃³×C₄
Frattini, cosocle	C₂×C₃²	C₂²×C₃³
socle	C₂²×C₃²	C₂×C₃³
prime radical	C₂²×C₃³	C₂×C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ G

456 conjugacy classes of subgroups; only the 288 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 288 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) = 42
transitive degree	μ_t(G) = 648

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2)(7,8,10,9,12,14,13,11,15)(16,17,20)(18,22,28)(19,23,27)(21,29,26)(24,31,36)(25,32,35)(30,37,34)(33,39,41)(38,42,40);(3,4,5,6)(17,21,23)(20,27,29)(22,30,32)(28,35,37)(31,38,39)(36,41,42);(8,11,12)(10,14,15)(16,18,24,19,25,33,26,34,40)(17,22,31,23,32,39,21,30,38)(20,28,36,27,35,41,29,37,42)

Conjugacy classes & character table

408 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1944, 2174);