C₂×#8195107 #72298997

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

Aliases

direct product

C₂×#8195107

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

54709 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297217
maximal normal subgroup of	C₂×#72297217
maximal quotient of	C₂×#72297599

GAP

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

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

C2×#8195107 #72298997