C₃×#2211 #72926

order	384
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#2211

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃
prime radical	C₂×C₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×C₈ ◁ G

158 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 39
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 39 points:

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

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#2211, #10949453, #10949469, …
maximal subgroup of	C₂×C₃×#2211, C₃×#33579, C₃×#33581, …
maximal normal subgroup of	C₂×C₃×#2211, C₃×#33579, C₃×#33581, …
maximal quotient of	C₃×#12419, C₃×#12423, C₃×#12429, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5502);