C₃×#2113 #72828

order	384
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#2113

Subgroups

	subgroup	G/H
center	C₂×C₃×C₈	C₂³
commutator	C₂²	C₂²×C₃×C₈
Frattini, cosocle	C₂×C₈	C₂³×C₃
socle	C₂²×C₃	C₂²×C₈
prime radical	C₂×C₃×#41	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ G

228 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#10682194, #10682197, #10682254, …
maximal subgroup of	C₃×#7207, C₃×#7208, C₃×#7227, …
maximal normal subgroup of	C₃×#7207, C₃×#7208, C₃×#7227, …
maximal quotient of	C₃×#11419, C₃×#11441, C₃×#11461, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5404);