C₂×C₃×#1840 #10639378

order	768
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₃×#1840

Subgroups

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

894 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 462 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) = 37
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 4 generators on 37 points:

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

Conjugacy classes & character table

264 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13759);