C₂×#1303195 #72337942

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#1303195

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

40189 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)(7,8,9,10)(11,12,16,17)(13,18,14,15)(19,20)(21,22)(23,24)(25,26);(7,8,9,10)(12,17)(15,18)(19,21)(20,22)(23,26)(24,25);(7,8)(9,10)(11,13,16,14)(12,18,17,15)(19,22,25,23)(20,21,26,24);(4,6)(7,8)(9,10)(11,14,16,13)(12,15,17,18)(19,23,25,22)(20,21,26,24);(1,2);(3,4)(5,6)(11,15)(12,13)(14,17)(16,18)(19,24,25,21)(20,23,26,22)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232318, C₂×#72232339, C₂×#72232296
maximal normal subgroup of	C₂×#72232318, C₂×#72232339, C₂×#72232296

GAP

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

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

C2×#1303195 #72337942