C₂×#7736813 #72331923

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#7736813

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

56697 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232288
maximal normal subgroup of	C₂×#72232288
maximal quotient of	C₂×#72323413

GAP

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

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

C2×#7736813 #72331923