C₂×#4423121 #72324461

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#4423121

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

52521 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232036, C₂×#72232041
maximal normal subgroup of	C₂×#72232036, C₂×#72232041

GAP

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

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

C2×#4423121 #72324461