C₂×#7630948 #72307153

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

Aliases

direct product

C₂×#7630948

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

107397 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)(9,13,14,12)(15,16)(17,18)(19,21)(20,22);(7,9)(8,12)(10,13)(11,14)(15,17,19,20)(16,18,21,22);(4,6)(9,14)(12,13)(15,16,19,21)(17,22,20,18);(3,4,5,6)(7,10,11,8)(9,12,14,13)(15,16)(17,18)(19,21)(20,22);(1,2);(15,18)(16,20)(17,21)(19,22);(7,9)(8,12)(10,13)(11,14)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236479, C₂×#72252111, C₂×#72310661, …
maximal normal subgroup of	C₂×#72236479, C₂×#72252111, C₂×#72310661, …

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

C2×#7630948 #72307153