C₂×C₃×#6913 #13090414

order	1458
nilpotency class	3
isoclinism	isoclinic class of #6909
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁴ · Schur multiplier (H₂) = C₃⁶

Aliases

direct product

C₂×C₃×#6913

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₂×C₃⁴
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₉ ◁ G

992 conjugacy classes of subgroups; only the 476 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 476 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) = 32
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 4 generators on 32 points:

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×C₃⋊(#6913⋊C₂)
maximal subgroup of	C₂×C₃×#10624934, C₂×#14554867, C₂×C₃⋊(#6913⋊C₂)
maximal normal subgroup of	C₂×C₃×#10624934, C₂×#14554867, C₂×C₃⋊(#6913⋊C₂)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1458, 488);