C₂×#72327577 #72331673

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁷

Aliases

direct product

C₂×#72327577

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

1763432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₂×#72327629

GAP

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

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

C2×#72327577 #72331673