C₂×#640970 #72334542

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

Aliases

direct product

C₂×#640970

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

73345 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)(15,16,19,20)(17,21,22,18);(4,6)(8,10)(12,13)(17,22)(18,21);(3,4)(5,6)(7,8)(9,10)(11,13,14,12)(15,17)(16,18)(19,22)(20,21);(1,2);(8,10)(15,16)(17,21)(18,22)(19,20);(3,4)(5,6)(7,8)(9,10)(15,18,19,21)(16,17,20,22)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72334296, C₂×#72303304
maximal normal subgroup of	C₂×#72334296, C₂×#72303304
maximal quotient of	C₂×#72334313

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

C2×#640970 #72334542