C₂×#5926 #87494

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂×2₋¹⁺⁴ ⋊ D₃

	subgroup	G/H
center, socle	C₂²	C₂²×S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂⁴
Frattini	C₂	C₂³×S₄
Fitting	C₂×2₋¹⁺⁴	D₃
hypercenter	C₂×D₄	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×D₄

797 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 797 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 32

A minimal faithful permutation representation — 4 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20070);