C₂×#7721789 #72330614

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7721789

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

108549 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232294
maximal normal subgroup of	C₂×#72232294
maximal quotient of	C₂×#72261284, C₂×#72261595, C₂×#72323285

GAP

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

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

C2×#7721789 #72330614