D₃×D₇×SL(2,𝔽₃) #72269501

order	2016
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

Fitting, derived	C₃×C₇×Q₈ ⋊ C₂²×C₃
Sylow	C₇ ⋊ C₂×D₃×SL(2,𝔽₃)

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×A₄×D₇
commutator, Fitting, cosocle, hypocenter	C₃×C₇×Q₈	C₂²×C₃
socle	C₂×C₃×C₇	C₂²×A₄
LCS	G ▷ C₃×C₇×Q₈
UCS	C₁ ◁ C₂

258 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 2 generators on 18 points:

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

maximal subgroup of	D₃×D₇×#997, D₃×SL(2,𝔽₃)×AGL(1,𝔽₇)
maximal normal subgroup of	D₃×SL(2,𝔽₃)×AGL(1,𝔽₇)

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

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