C₂×#1409003 #72320362

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄³

Aliases

direct product

C₂×#1409003

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

65945 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72320249, C₂²×#1409003
maximal normal subgroup of	C₂×#72320249, C₂²×#1409003
maximal quotient of	C₂×#72278310, C₂²×#1409003

GAP

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

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

C2×#1409003 #72320362