C₂×#8093620 #72331676

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

Aliases

direct product

C₂×#8093620

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

75365 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)(5,9)(6,8)(7,10)(11,12)(13,17)(14,15)(16,18)(19,20,23,24)(21,25,26,22);(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(20,24)(22,25);(3,6)(4,8)(5,10)(7,9)(11,14)(12,15)(13,18)(16,17);(1,2);(11,14)(12,13)(15,18)(16,17)(19,21)(20,22)(23,26)(24,25);(3,6)(4,5)(7,9)(8,10)(11,14)(12,13)(15,18)(16,17);(3,5)(4,9)(6,10)(7,8)(11,15,16,13)(12,17,18,14)(19,22)(20,21)(23,25)(24,26)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72241227
maximal normal subgroup of	C₂×#72241227
maximal quotient of	C₂×#72327582

GAP

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

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

C2×#8093620 #72331676