C₂×#7859844 #72279014

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

Aliases

direct product

C₂×#7859844

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

72393 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)(6,8)(7,10)(11,12)(13,16)(14,17)(15,18)(21,25)(23,26);(3,4,6,8)(5,9,10,7)(11,12,15,18)(13,16,14,17);(11,13,15,14)(12,16,18,17)(19,20,22,24)(21,23,25,26);(13,14)(16,17)(21,25)(23,26);(3,5,6,10)(4,7,8,9)(11,14,15,13)(12,17,18,16)(19,21)(20,23)(22,25)(24,26);(3,4)(5,9)(6,8)(7,10)(11,12)(13,17)(14,16)(15,18)(19,20)(21,23)(22,24)(25,26);(1,2)

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232276, C₂×#72232288
maximal normal subgroup of	C₂×#72232276, C₂×#72232288
maximal quotient of	C₂×#72323251, C₂×#72323252, C₂×#72323249

GAP

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

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

C2×#7859844 #72279014