C₂×#7744850 #72302197

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

Aliases

direct product

C₂×#7744850

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

53365 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72279634
maximal normal subgroup of	C₂×#72279634
maximal quotient of	C₂×#72280318, C₂×#72279775, C₂×#72280169

GAP

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

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

C2×#7744850 #72302197