C₂×#2123 #33346

order	256
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#2123

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#99
commutator	C₂³	C₂³×C₄
Frattini, cosocle	C₂×D₄	C₂⁴
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁

607 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 607 conjugacy classes of subgroups — too many to draw. Only its 183 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4)(5,13)(6,10)(7,20,21,11)(8,12)(9,15,23,30)(14,27)(16,28)(17,31)(18,33,34,25)(19,26)(22,29,24,32);(3,5,8,16)(4,9,12,23)(6,17,19,14)(7,15,21,30)(10,24,26,22)(11,13,20,28)(18,32,34,29)(25,31,33,27);(3,6,8,19)(4,10,12,26)(5,14,16,17)(7,18,21,34)(9,22,23,24)(11,25,20,33)(13,27,28,31)(15,29,30,32)

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#146612, #146617, #146681, …
maximal normal subgroup of	#146612, #146617, #146681, …
maximal quotient of	C₂×#11428, C₂×#11454, C₂×#11474, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26334);