C₂×#72323646 #72339377

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#72323646

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁵×#120	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2735320 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72337376
maximal normal subgroup of	C₂×#72337376

GAP

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

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

C2×#72323646 #72339377