C₂×#72297328 #72298968

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁵

Aliases

direct product

C₂×#72297328

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁷
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 16032 conjugacy classes of subgroups; only the 418552 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 26 points:

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

Conjugacy classes & character table

332 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₂×#72297602

GAP

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

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

C2×#72297328 #72298968