C₂×#8357779 #72329309

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

Aliases

direct product

C₂×#8357779

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

72505 conjugacy classes of subgroups.

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232285, C₂×#72232275
maximal normal subgroup of	C₂×#72232285, C₂×#72232275
maximal quotient of	C₂×#72232296

GAP

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

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

C2×#8357779 #72329309