C₂×#7819612 #72331806

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

Aliases

direct product

C₂×#7819612

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

57001 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)(5,9)(6,10)(7,8)(12,15)(16,17)(19,20,22,24)(21,23,26,25);(11,12,14,15)(13,16,18,17)(19,21)(20,23)(22,26)(24,25);(4,8)(6,9)(11,12)(13,17)(14,15)(16,18);(11,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,24)(23,26);(1,2);(3,5)(4,9)(6,8)(7,10)(11,12)(13,17)(14,15)(16,18);(3,6,7,9)(4,10,8,5)(11,13,14,18)(12,16,15,17)(19,21)(20,23)(22,26)(24,25)

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×#72323421

GAP

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

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

C2×#7819612 #72331806