C₂×#130234 #72280731

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#130234

Subgroups

	subgroup	G/H
center, socle	C₂²	#8350
commutator	C₂×M₁₆	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
prime radical	C₂×#62328	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

4386 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231624, C₂×#72231712, #72313820, …
maximal normal subgroup of	C₂×#72231624, C₂×#72231712, #72313820, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C2×#130234 #72280731