C₂×#8346905 #72331126

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

Aliases

direct product

C₂×#8346905

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

67885 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232290, C₂×#72232275
maximal normal subgroup of	C₂×#72232290, C₂×#72232275
maximal quotient of	C₂×#72232314

GAP

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

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

C2×#8346905 #72331126