C₃×#32948 #10651555

order	768
nilpotency class	3
isoclinism	isoclinic class of #32947
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₃×#32948

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#2847
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ G

1334 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 290 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) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 25936);