C₂×#617165 #72334684

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#617165

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

69357 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72330222, C₂×#72297882, C₂×#72331394, …
maximal normal subgroup of	C₂×#72330222, C₂×#72297882, C₂×#72331394, …
maximal quotient of	C₂×#72334395, C₂×#72298450

GAP

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

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

C2×#617165 #72334684