C₃×#72275713 #72275847

order	6048
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, hypercenter	C₂×C₃	C₃⋊SO(3,𝔽₇)
commutator, cosocle, hypocenter	C₃×SL(2,𝔽₇)	C₂×C₃
Frattini	C₂	C₃×C₃⋊SO(3,𝔽₇)
Fitting, socle	C₂×C₃²	SO(3,𝔽₇)
LCS	G ▷ C₃×SL(2,𝔽₇)
UCS	C₁ ◁ C₂×C₃

219 conjugacy classes of subgroups; only the 14 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 14 normal subgroups are shown.

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

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 22 points:

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

maximal subgroup of	C₃×D₃×#10621462
maximal normal subgroup of	C₃×D₃×#10621462

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

gap> G := Group((1,3)(7,20,8,10,18,17)(9,22,14,21,11,19)(15,16), (2,3)(4,5,6)(7,18,17,15,10,12,20,8,19,21,11,16,14,13,9,22));