C₂×C₃×AGL(1,ℤ/32) #72255395

order	3072
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₃×AGL(1,ℤ/32)

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#12067
commutator	C₁₆	C₂³×C₃×C₈
Frattini, cosocle	#346	C₂⁴×C₃
prime radical	C₂×C₃×#2263	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ C₂×C₃×#1313 ◁ G

1298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 37 points:

(3,4,5,9,6,10,13,22,7,11,14,23,16,25,29,19,8,12,15,21,17,26,28,33,18,24,30,20,31,34,27,32)(35,36,37);(1,2);(4,9,11,20,12,21,24,23)(5,13,15,28)(6,16)(7,18)(10,19,25,22,26,32,34,33)(14,27,30,29)(17,31);(4,10,24,34,12,26,11,25)(5,14,15,30)(6,17)(9,19,23,33,21,32,20,22)(13,27,28,29)(16,31)

Conjugacy classes & character table

282 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	C₂×D₃×AGL(1,ℤ/32)
maximal subgroup of	C₂×D₃×AGL(1,ℤ/32)
maximal normal subgroup of	C₂×D₃×AGL(1,ℤ/32)

GAP

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

gap> G := Group((3,4,5,9,6,10,13,22,7,11,14,23,16,25,29,19,8,12,15,21,17,26,28,33,18,24,30,20,31,34,27,32)(35,36,37), (1,2), (4,9,11,20,12,21,24,23)(5,13,15,28)(6,16)(7,18)(10,19,25,22,26,32,34,33)(14,27,30,29)(17,31), (4,10,24,34,12,26,11,25)(5,14,15,30)(6,17)(9,19,23,33,21,32,20,22)(13,27,28,29)(16,31));

C2×C3×AGL(1,ℤ/32) #72255395