C₃×#7622637 #72228228

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₃×#7622637

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂²×#442
commutator	C₈	C₂⁴×C₃×C₄
Frattini, cosocle	C₂×C₈	C₂⁵×C₃
prime radical	C₃×#62636	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×#41 ◁ C₃×#2875 ◁ G

3826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 35 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	D₃×#7622637
maximal subgroup of	C₃×#72228227, D₃×#7622637
maximal normal subgroup of	D₃×#7622637
maximal quotient of	C₃×#72228227

GAP

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

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

C3×#7622637 #72228228