C₂×C₃×#2198 #10652152

order	768
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×C₃×#2198

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂⁴×C₃
prime radical	C₂×C₃×#453	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃ ◁ C₂²×C₃×D₄ ◁ G

1244 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1244 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) = 21
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 21 points:

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

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, 26533);