C₂×#72277950 #72296431

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#72277950

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 23652 conjugacy classes of subgroups; only the 38096 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72298104
maximal normal subgroup of	C₂×#72298104
maximal quotient of	C₂×#72278229

GAP

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

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

C2×#72277950 #72296431