C₂×#7664938 #72298412

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂×#7664938

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

89945 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)(9,12)(10,14)(17,21)(19,22);(3,4)(5,6)(7,8,11,13)(9,10,12,14)(15,16,18,20)(17,19,21,22);(7,9,11,12)(8,10,13,14)(15,16)(17,19)(18,20)(21,22);(7,9,11,12)(8,10,13,14)(15,17,18,21)(16,19,20,22);(7,9)(8,10)(11,12)(13,14);(7,10)(8,12)(9,13)(11,14);(1,2)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72234453
maximal normal subgroup of	C₂×#72234453
maximal quotient of	C₂×#72297594, C₂×#72298097

GAP

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

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

C2×#7664938 #72298412