C₂×#7745123 #72302196

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

Aliases

direct product

C₂×#7745123

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

At least 9072 conjugacy classes of subgroups; only the 29773 normal subgroups are shown.

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)(5,6)(7,9)(8,10)(11,12)(13,18)(14,16)(15,17)(19,20,23,21)(22,24,26,25);(11,12,15,17)(13,14,16,18)(22,26)(24,25);(5,9)(6,7)(11,12)(13,18)(14,16)(15,17);(3,5)(4,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,21,23,20)(22,25,26,24);(3,6,8,7)(4,9,10,5)(11,13,15,16)(12,14,17,18)(19,22)(20,24)(21,25)(23,26);(3,7,8,6)(4,9,10,5)(11,12)(13,14)(15,17)(16,18);(3,4)(5,7)(6,9)(8,10)(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×#72280257, C₂×#72279836, C₂×#72279926, …

GAP

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

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

C2×#7745123 #72302196