D₃×#491469 #72225931

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₈

Aliases

direct product

D₃×#491469

Splittings

Fitting	C₃×#491469 ⋊ C₂
Sylow	C₃ ⋊ C₂×#491469

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×#493
commutator, cosocle	C₃×C₄×C₈	C₂⁵
Frattini	C₄×C₈	C₂⁴×D₃
Fitting	C₃×#491469	C₂
socle	C₂²×C₃	C₂²×#493
hypercenter	#491469	D₃
hypocenter	C₃	C₂×#491469
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #491469

10458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 5 generators on 27 points:

(1,2,3)(12,13,19,27,15,20,14,17,16,18,21,26,22,25,23,24);(4,5,9,11,7,10,6,8);(13,18)(17,24)(20,25)(26,27);(2,3);(5,8)(6,9)(10,11)(13,17)(14,21)(15,22)(18,24)(19,23)(20,27)(25,26)

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)(12,13,19,27,15,20,14,17,16,18,21,26,22,25,23,24), (4,5,9,11,7,10,6,8), (13,18)(17,24)(20,25)(26,27), (2,3), (5,8)(6,9)(10,11)(13,17)(14,21)(15,22)(18,24)(19,23)(20,27)(25,26));

D3×#491469 #72225931