C₂×#612794 #72334533

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#612794

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

71881 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₄×#2849, C₂×#72334291, C₂×#72303304, …
maximal normal subgroup of	C₂×D₄×#2849, C₂×#72334291, C₂×#72303304, …
maximal quotient of	C₂×#72334309, C₂×#72334403

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

C2×#612794 #72334533