C₂×#7783989 #72323792

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂×#7783989

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

98653 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)(5,6)(7,8)(9,13)(10,12)(11,14)(15,16)(17,18)(19,20)(21,22);(3,4)(5,6)(7,8,10,12)(9,11,14,13)(15,17,20,22)(16,21,19,18);(3,4)(5,6)(9,14)(11,13)(15,18,20,21)(16,22,19,17);(4,6)(15,18)(16,22)(17,19)(20,21);(1,2);(3,4,5,6)(7,9,10,14)(8,11,12,13)(15,17)(16,18)(19,21)(20,22);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,19,20,16)(17,18,22,21)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232031, C₂²×#7783989
maximal normal subgroup of	C₂×#72232031, C₂²×#7783989
maximal quotient of	C₂×#72323263, C₂×#72232034, C₂×#72232171, …

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

C2×#7783989 #72323792