C₃×#33758 #10652365

order	768
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃×#33758

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ C₂³×C₃×C₈ ◁ G

688 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26746);