C₃×#33755 #10652362

order	768
nilpotency class	4
isoclinism	isoclinic class of #33752
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#33755

Subgroups

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

590 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 214 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) = 43
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 43 points:

(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13,17,21)(14,22,25,41)(15,26,30,40)(16,31,28,20)(18,33,35,23)(19,36,38,29)(24,42,32,37)(27,39,43,34);(4,6)(5,8)(7,11)(9,10);(6,11)(8,10)(12,14,17,25)(13,18,21,35)(15,27,30,43)(16,32,28,24)(19,37,38,42)(20,39,31,34)(22,40,41,26)(23,36,33,29);(5,9)(8,10)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26743);