C₃×#72209884 #72276696

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#72209884

Splittings

Fitting	C₃×#15947 ⋊ D₃
derived	#11709313 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center	C₂×C₃	#11714279
commutator, cosocle, hypocenter	#11709313	C₂×C₃
Frattini	C₂⁴	C₃×#1020
Fitting	C₃×#15947	D₃
socle	C₂³×C₃	#5934
hypercenter	C₂²×C₃	#85559
LCS	G ▷ #11709313
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃

2395 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 2395 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 27
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 27 points:

(1,2,3)(4,10,8,21)(5,7)(6,22,24,12,15,26,14,18)(9,25,27,17)(11,16)(13,20,23,19);(4,15,13)(5,9,12)(6,17,8)(7,24,23)(10,19,22)(11,20,18)(14,25,16)(21,27,26)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃×#72209884
maximal normal subgroup of	C₂×C₃×#72209884
maximal quotient of	C₂×C₃×#72209884

GAP

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

gap> G := Group((1,2,3)(4,10,8,21)(5,7)(6,22,24,12,15,26,14,18)(9,25,27,17)(11,16)(13,20,23,19), (4,15,13)(5,9,12)(6,17,8)(7,24,23)(10,19,22)(11,20,18)(14,25,16)(21,27,26));