C₂×#7630946 #72307161

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

Aliases

direct product

C₂×#7630946

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

108477 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236479, C₂×#72252578, C₂×#72252568, …
maximal normal subgroup of	C₂×#72236479, C₂×#72252578, C₂×#72252568, …
maximal quotient of	C₂×#72335120, C₂×#72335132

GAP

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

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

C2×#7630946 #72307161