C₂×#7850075 #72323782

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

Aliases

direct product

C₂×#7850075

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 7022 conjugacy classes of subgroups; only the 31973 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:

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

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232031, C₂×#72232032
maximal normal subgroup of	C₂×#72232031, C₂×#72232032
maximal quotient of	C₂×#72323254, C₂×#72261233, C₂×#72232052, …

GAP

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

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

C2×#7850075 #72323782