C₂×#7659308 #72331975

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

Aliases

direct product

C₂×#7659308

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

59669 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232288
maximal normal subgroup of	C₂×#72232288
maximal quotient of	C₂×#72323399

GAP

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

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

C2×#7659308 #72331975