C₂×#8358079 #72337790

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

Aliases

direct product

C₂×#8358079

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

67933 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)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20,23,22)(21,25,26,24);(3,4,6,8)(5,9,10,7)(11,12)(13,17)(14,16)(15,18)(19,21,23,26)(20,24,22,25);(3,4)(5,7)(6,8)(9,10)(11,12,14,16)(13,17,18,15)(19,21,23,26)(20,25,22,24);(5,10)(7,9)(13,18)(15,17)(19,21)(20,24)(22,25)(23,26);(1,2);(3,4)(5,7)(6,8)(9,10)(11,12)(13,17)(14,16)(15,18)(19,20,23,22)(21,25,26,24);(3,5,6,10)(4,7,8,9)(11,13,14,18)(12,15,16,17)(19,22,23,20)(21,24,26,25)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232275
maximal normal subgroup of	C₂×#72232275
maximal quotient of	C₂×#72232305

GAP

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

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

C2×#8358079 #72337790