C₂×#72334964 #72339998

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#72334964

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁹	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

8158728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72334969
maximal normal subgroup of	C₂×#72334969

GAP

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

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

C2×#72334964 #72339998