C₂×#7792785 #72293995

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

Aliases

direct product

C₂×#7792785

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 10824 conjugacy classes of subgroups; only the 31901 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72238224, C₂×#72232067
maximal normal subgroup of	C₂×#72238224, C₂×#72232067
maximal quotient of	C₂×#72250236, C₂×#72293921, C₂×#72293937, …

GAP

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

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

C2×#7792785 #72293995