C₂×#719098 #72334532

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

Aliases

direct product

C₂×#719098

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

73853 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)(12,14)(15,16,19,22)(17,20,18,21);(4,6)(8,10)(11,12)(13,14)(17,18)(20,21);(3,4)(5,6)(7,8,9,10)(12,14)(15,17,19,18)(16,20,22,21);(8,10)(11,12)(13,14)(15,16)(17,21)(18,20)(19,22);(1,2);(3,4)(5,6)(7,8)(9,10)(15,18,19,17)(16,21,22,20)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72334296, C₂×#72327596, C₂×#72327589
maximal normal subgroup of	C₂×#72334296, C₂×#72327596, C₂×#72327589
maximal quotient of	C₂×#72316973

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

C2×#719098 #72334532