C₂×#7627266 #72323779

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7627266

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#3484	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

99505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72232031, C₂×#72232293
maximal normal subgroup of	C₂×#72232031, C₂×#72232293
maximal quotient of	C₂×#72323253, C₂×#72232057, C₂×#72232121, …

GAP

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

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

C2×#7627266 #72323779