C₂×#8689971 #72331982

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

Aliases

direct product

C₂×#8689971

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

57519 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:

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

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232288
maximal normal subgroup of	C₂×#72232288
maximal quotient of	C₂×#72323397

GAP

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

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

C2×#8689971 #72331982