C₂×#2822351 #72294845

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#2822351

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

48923 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

(3,4,5,6)(11,12,15,16)(13,14,17,18)(19,20,24,25)(21,26,22,23);(3,4,5,6)(12,16)(14,18)(20,25)(21,22);(3,4)(5,6)(8,10)(13,17)(14,18)(19,21,24,22)(20,23,25,26);(3,4,5,6)(7,8)(9,10)(11,13,15,17)(12,14,16,18)(19,22,24,21)(20,23,25,26);(3,4)(5,6)(11,14)(12,13)(15,18)(16,17)(19,23)(20,22)(21,25)(24,26);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293915, C₂×#72231349, C₂²×#2822351, …
maximal normal subgroup of	C₂×#72293915, C₂×#72231349, C₂²×#2822351, …
maximal quotient of	C₂×#72293951, C₂²×#2822351, C₂×#72278150

GAP

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

gap> G := Group((3,4,5,6)(11,12,15,16)(13,14,17,18)(19,20,24,25)(21,26,22,23), (3,4,5,6)(12,16)(14,18)(20,25)(21,22), (3,4)(5,6)(8,10)(13,17)(14,18)(19,21,24,22)(20,23,25,26), (3,4,5,6)(7,8)(9,10)(11,13,15,17)(12,14,16,18)(19,22,24,21)(20,23,25,26), (3,4)(5,6)(11,14)(12,13)(15,18)(16,17)(19,23)(20,22)(21,25)(24,26), (1,2));

C2×#2822351 #72294845