C₂×#664642 #72334516

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#664642

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

78197 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)(12,13)(15,16)(17,19)(18,21)(20,22);(7,8)(9,10)(11,12)(13,14)(16,17)(18,22);(4,5)(8,9)(11,12)(13,14)(15,16,19,17)(18,21,22,20);(4,5)(8,9)(12,13)(15,17,19,16)(18,20,22,21);(1,2);(3,5,6,4)(7,9,10,8)(11,13,14,12)(15,18)(16,20)(17,21)(19,22)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72334295, C₂×D₄×#2405, C₂×#72334297
maximal normal subgroup of	C₂×#72334295, C₂×D₄×#2405, C₂×#72334297
maximal quotient of	C₂×#72316956

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

C2×#664642 #72334516