C₂×#3026 #60395

order	256
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	Aut(G) order 98304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×#3026

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#120
commutator, Frattini, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₂ ▷ C₁

2659 conjugacy classes of subgroups; only the 559 normal subgroups are shown.

normal subgroups only

This group has 2659 conjugacy classes of subgroups — too many to draw. Only its 559 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) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#3026⋊C₃, #72227797, C₂×#11714182, …
maximal subgroup of	#147752, #364500, #364545, …
maximal normal subgroup of	#147752, #364500, #364545, …
maximal quotient of	C₂×#18284, C₂×#18285, C₂×#18286, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53383);