C₂×#7630947 #72307147

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

Aliases

direct product

C₂×#7630947

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

108413 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236479, C₂×#72235992, #72314957, …
maximal normal subgroup of	C₂×#72236479, C₂×#72235992, #72314957, …

GAP

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

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

C2×#7630947 #72307147