C₂×#7901701 #72298990

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

Aliases

direct product

C₂×#7901701

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

At least 6934 conjugacy classes of subgroups; only the 31893 normal subgroups are shown.

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297863, C₂×#72232032, C₂×#72232070
maximal normal subgroup of	C₂×#72297863, C₂×#72232032, C₂×#72232070
maximal quotient of	C₂×#72298085, C₂×#72232047, C₂×#72232103

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

C2×#7901701 #72298990