C₂×#857⋊C₂ #85373

order	384
isoclinism	isoclinic class of #857⋊C₂
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product	C₂×#857⋊C₂
unambiguous semidirect product	#857⋊C₂²

Fitting	C₂×#127 ⋊ D₃
derived	C₄²⋊C₃ ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂	#857⋊C₂
commutator, cosocle, hypocenter	C₄²⋊C₃	C₂³
Frattini	C₂²	C₂²×S₄
Fitting	C₂×#127	D₃
socle	C₂³	C₂×S₄
LCS	G ▷ C₄²⋊C₃
UCS	C₁ ◁ C₂

392 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.

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

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 3 generators on 12 points:

(1,7,12,5,6,10)(2,9,11,3,4,8);(4,8)(6,10)(7,11)(9,12);(1,2)(3,5)(4,6)(7,9)(8,10)(11,12)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 17949);