C₃×#136363 #72228262

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#136363

Subgroups

	subgroup	G/H
center	C₃×C₄	#1629
commutator	C₄×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³×C₃
socle	C₂×C₃	#8886
prime radical	C₂×C₃×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ G

666 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72214178
maximal normal subgroup of	C₃×#72214178

GAP

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

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

C3×#136363 #72228262