D₃×#135792 #72226185

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₈

Aliases

direct product

D₃×#135792

Splittings

Fitting	C₃×#135792 ⋊ C₂
Sylow	C₃ ⋊ C₂×#135792

Subgroups

	subgroup	G/H
center	C₂²	D₃×#1844
commutator	C₃×C₄×C₈	C₂³×C₄
Frattini	C₂×C₄×C₈	C₂³×D₃
Fitting	C₃×#135792	C₂
socle	C₂²×C₃	C₂×#1844
prime radical	C₂³×D₃	#498
cosocle	C₂×C₃×C₄×C₈	C₂⁴
hypercenter	#135792	D₃
hypocenter	C₃	C₂×#135792
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #135792

2254 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 768

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

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

Conjugacy classes & character table

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

GAP

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

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

D3×#135792 #72226185