D₃×#347042 #72231423

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

D₃×#347042

Splittings

Fitting	C₃×#347042 ⋊ C₂
Sylow	C₃ ⋊ C₂×#347042

Subgroups

	subgroup	G/H
center	C₂²	D₃×#2846
commutator, cosocle	C₂⁵×C₃	C₂⁵
Frattini	C₂⁵	C₂⁴×D₃
Fitting	C₃×#347042	C₂
socle	C₂²×C₃	C₂×#2846
hypercenter	#347042	D₃
hypocenter	C₃	C₂×#347042
LCS	G ▷ C₂⁵×C₃ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #347042

70142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 5 generators on 19 points:

(1,2,3)(12,14)(13,16)(15,19)(17,18);(4,6)(5,8)(7,10)(9,11);(4,5)(6,8)(7,9)(10,11)(12,13)(14,18)(15,17)(16,19);(2,3);(8,11)(16,18)

Conjugacy classes & character table

177 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72209641, D₃×B(5,𝔽₂), C₂×D₃×#347042
maximal normal subgroup of	D₃×B(5,𝔽₂), C₂×D₃×#347042
maximal quotient of	C₂×D₃×#347042
automorphism group of	C₂×C₄⋊Dic₃

GAP

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

gap> G := Group((1,2,3)(12,14)(13,16)(15,19)(17,18), (4,6)(5,8)(7,10)(9,11), (4,5)(6,8)(7,9)(10,11)(12,13)(14,18)(15,17)(16,19), (2,3), (8,11)(16,18));

D3×#347042 #72231423