C₂×#7837395 #72323788

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

Aliases

direct product

C₂×#7837395

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 6800 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72238224, C₂×#72232070
maximal normal subgroup of	C₂×#72238224, C₂×#72232070
maximal quotient of	C₂×#72261233, C₂×#72297746, C₂×#72232126, …

GAP

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

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

C2×#7837395 #72323788