C₂×#3333147 #72332677

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

Aliases

direct product

C₂×#3333147

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

31743 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323396, C₂×#72323408, C₂×#72323388
maximal normal subgroup of	C₂×#72323396, C₂×#72323408, C₂×#72323388

GAP

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

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

C2×#3333147 #72332677