C₂×#7729198 #72324744

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

Aliases

direct product

C₂×#7729198

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

97885 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232070
maximal normal subgroup of	C₂×#72232070
maximal quotient of	C₂×#72232116

GAP

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

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

C2×#7729198 #72324744