C₃×#493226 #72226273

order	1536
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#493226

Subgroups

	subgroup	G/H
center	C₃×C₄	C₄²×D₄
commutator	C₂×C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴×C₃
socle	C₂×C₃	D₄×#97
prime radical	C₂×C₃×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×#347 ◁ G

1328 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×#72214092
maximal normal subgroup of	C₃×#72214092

GAP

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

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

C3×#493226 #72226273