C₂×C₃×#12755 #72254279

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×C₃×#12755

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#1790
commutator	C₂×D₄	C₂³×C₃×C₄
Frattini, cosocle	C₂×#41	C₂⁴×C₃
prime radical	C₂×C₃×#3537	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×#41 ◁ G

5598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×D₃×#12755, #72254320
maximal subgroup of	C₃×#72215601, C₂×D₃×#12755, #72254320
maximal normal subgroup of	C₃×#72215601, C₂×D₃×#12755, #72254320

GAP

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

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

C2×C3×#12755 #72254279