C₂×#783603 #72337988

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#783603

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

40091 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232357, C₂×#72232295, C₂×#72232296
maximal normal subgroup of	C₂×#72232357, C₂×#72232295, C₂×#72232296

GAP

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

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

C2×#783603 #72337988