D₃×#11715828 #72224558

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

D₃×#11715828

Splittings

Fitting

C₃×H(7,𝔽₂) ⋊ C₂×D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×D₃×#1020
commutator, cosocle, hypocenter	C₃×#997	C₂⁴
Fitting	C₃×H(7,𝔽₂)	C₂×D₃
socle	C₂×C₃	C₂³×#1020
hypercenter	D₄	D₃×#1020
LCS	G ▷ C₃×#997
UCS	C₁ ◁ C₂ ◁ D₄

At least 10088 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72217027
maximal normal subgroup of	D₃×#72217027
holomorph of	C₃×#41

GAP

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

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

D3×#11715828 #72224558