C₃×#7385 #10625992

order	768
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7385

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#357
commutator	C₁₆	C₂×C₃×C₈
Frattini, cosocle	C₄×C₁₆	C₂²×C₃
prime radical	C₂×C₃×D₁₆	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×C₈ ◁ C₃×C₄×C₁₆ ◁ G

136 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 43
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 2 generators on 43 points:

(1,2,3)(4,5,6,8,7,9,10,11)(12,13,15,21,17,23,18,34)(14,24,27,41,29,19,30,35)(16,31,26,38,33,39,32,22)(20,37,36,25,40,43,42,28);(13,19)(14,25)(15,18)(16,32)(20,38)(21,41)(22,40)(23,24)(26,33)(27,37)(28,29)(30,43)(31,36)(34,35)(39,42)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#7385, #72245006
maximal subgroup of	C₃×#151085, D₃×#7385, #72245006, …
maximal normal subgroup of	C₃×#151085, D₃×#7385, #72245006, …
maximal quotient of	C₃²×#7385, C₃×#94614, C₃×#94613

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 373);