D₃×#504 #82011

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₃×#504

Splittings

Fitting	C₃×#504 ⋊ C₂
Sylow	C₃ ⋊ C₂×#504

Subgroups

	subgroup	G/H
center	C₁₆	C₂²×D₃
commutator, socle	C₂×C₃	C₂³×C₈
Frattini	C₈	C₂³×D₃
Fitting	C₃×#504	C₂
prime radical	D₃×#41	C₄
cosocle	C₃×C₈	C₂⁴
hypercenter	#504	D₃
hypocenter	C₃	C₂×#504
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₁₆ ◁ #504

368 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 211 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) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1401, #10715396, D₃×#2170, …
maximal normal subgroup of	D₃×#1401, #10715396, D₃×#2170, …
maximal quotient of	D₃×#2108, Dic₃×#504, #10947831, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14587);