C₂×#7626593 #72335168

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

Aliases

direct product

C₂×#7626593

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

110893 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72236218
maximal normal subgroup of	C₂×#72236218
maximal quotient of	C₂×#72335123, C₂×#72335136

GAP

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

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

C2×#7626593 #72335168