C₂×#8093107 #72241027

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

Aliases

direct product

C₂×#8093107

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

161337 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233384, #72241254, C₂×#72218920, …
maximal normal subgroup of	C₂×#72233384, #72241254, C₂×#72218920, …
maximal quotient of	#72241223, C₂×#72240189

GAP

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

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

C2×#8093107 #72241027