C₃×#153598 #72254994

order	1536
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#153598

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#13654
commutator	C₃₂	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₃₂	C₂³×C₃
prime radical	C₃×#13742	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×C₄×C₈ ◁ C₃×C₄×C₁₆ ◁ C₃×#1397 ◁ G

234 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 67
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 67 points:

(1,2,3,10,17,35,50,44,5,13,18,38,22,43,48,8,24,9,30,31,55,60,52,29,56,33,59,57,4,12,15,37,7,11,16,36,20,42,47,63,25,40,49,28,54,32,53,27,6,14,19,34,51,61,64,58,21,41,46,62,23,39,45,26)(65,66,67);(2,8,11,27)(3,15)(4,20)(5,21)(6,24)(9,26,14,37)(10,32,36,43)(12,34,39,31)(13,44,40,63)(16,45)(17,23)(18,52)(19,53)(22,51)(25,56)(28,35,38,42)(29,33,58,41)(30,48)(46,47)(49,64)(50,59)(54,55)(57,60,62,61);(2,9,11,14)(3,16)(8,26,27,37)(10,34,36,31)(12,32,39,43)(13,33,40,41)(15,45)(18,49)(19,30)(28,57,38,62)(29,44,58,63)(35,60,42,61)(46,59)(47,50)(48,53)(52,64)

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#153598, #72255028
maximal subgroup of	C₃×#72224652, D₃×#153598, #72255028
maximal normal subgroup of	C₃×#72224652, D₃×#153598, #72255028

GAP

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

gap> G := Group((1,2,3,10,17,35,50,44,5,13,18,38,22,43,48,8,24,9,30,31,55,60,52,29,56,33,59,57,4,12,15,37,7,11,16,36,20,42,47,63,25,40,49,28,54,32,53,27,6,14,19,34,51,61,64,58,21,41,46,62,23,39,45,26)(65,66,67), (2,8,11,27)(3,15)(4,20)(5,21)(6,24)(9,26,14,37)(10,32,36,43)(12,34,39,31)(13,44,40,63)(16,45)(17,23)(18,52)(19,53)(22,51)(25,56)(28,35,38,42)(29,33,58,41)(30,48)(46,47)(49,64)(50,59)(54,55)(57,60,62,61), (2,9,11,14)(3,16)(8,26,27,37)(10,34,36,31)(12,32,39,43)(13,33,40,41)(15,45)(18,49)(19,30)(28,57,38,62)(29,44,58,63)(35,60,42,61)(46,59)(47,50)(48,53)(52,64));