C₃×#11923520 #72245686

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²×C₃

solvable

Aliases

direct product

C₃×#11923520

Splittings

Fitting

C₃×#585 ⋊ C₃⋊D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄	A₄⋊S₄
commutator, cosocle, hypocenter	Ω⁺(4,𝔽₃)	C₂²×C₃
Frattini	C₂	C₂×C₃×A₄⋊S₄
Fitting	C₃×#585	C₃⋊D₃
socle	C₂×C₃	C₂×A₄⋊S₄
LCS	G ▷ Ω⁺(4,𝔽₃)
UCS	C₁ ◁ C₃×C₄

850 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 51
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 51 points:

(1,4,7,17)(2,9,12,25)(3,38,19,13)(5,20)(6,18)(8,48,27,21)(10,28)(11,26)(14,31,37,29)(15,32,35,36)(16,30,34,33)(22,41,47,39)(23,42,45,46)(24,40,44,43)(49,50,51);(1,8,33,12,3,21,7,24,13,2,16,43)(4,23,31,25,15,46,17,47,36,9,37,41)(5,27,32,28,19,39,20,44,29,10,34,42)(6,22,38,26,14,40,18,45,30,11,35,48);(4,5,6)(9,10,11)(14,15,19)(17,20,18)(22,23,27)(25,28,26)(29,30,36)(31,32,38)(34,35,37)(39,40,46)(41,42,48)(44,45,47)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72224794
maximal normal subgroup of	C₃×#72224794
maximal quotient of	#72245689

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,4,7,17)(2,9,12,25)(3,38,19,13)(5,20)(6,18)(8,48,27,21)(10,28)(11,26)(14,31,37,29)(15,32,35,36)(16,30,34,33)(22,41,47,39)(23,42,45,46)(24,40,44,43)(49,50,51), (1,8,33,12,3,21,7,24,13,2,16,43)(4,23,31,25,15,46,17,47,36,9,37,41)(5,27,32,28,19,39,20,44,29,10,34,42)(6,22,38,26,14,40,18,45,30,11,35,48), (4,5,6)(9,10,11)(14,15,19)(17,20,18)(22,23,27)(25,28,26)(29,30,36)(31,32,38)(34,35,37)(39,40,46)(41,42,48)(44,45,47));