C₂×#72232032 #72324375

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂×#72232032

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 33536 conjugacy classes of subgroups; only the 444832 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 8 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₂×#72232064

GAP

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

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

C2×#72232032 #72324375