C₂×C₃×#2201 #10652156

order	768
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×C₃×#2201

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂⁴×C₃
prime radical	C₂×C₃×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃ ◁ C₂²×C₃×D₄ ◁ G

1136 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 192

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

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26537);