C₂×#7633109 #72298415

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

Aliases

direct product

C₂×#7633109

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

89869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72234453
maximal normal subgroup of	C₂×#72234453
maximal quotient of	C₂×#72298086, C₂×#72297592

GAP

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

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

C2×#7633109 #72298415