C₂×#7696048 #72331027

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#7696048

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 9268 conjugacy classes of subgroups; only the 30005 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

214 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232290
maximal normal subgroup of	C₂×#72232290

GAP

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

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

C2×#7696048 #72331027