C₄×#433 #20464

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₄×#433

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂³
commutator	C₂²	C₂²×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂³	C₂³×C₄
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

485 conjugacy classes of subgroups; only the 303 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 303 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2,3,4);(5,6)(7,10)(8,9)(11,12)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(6,9)(10,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(5,7,8,11)(6,10,9,12)(15,21)(18,24)(22,27)(25,28)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×#433, #183749, #183750, …
maximal normal subgroup of	C₈×#433, #183749, #183750, …
maximal quotient of	C₈×#433, #183749, #183750, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13452);