C₂×#8126749 #72324815

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

Aliases

direct product

C₂×#8126749

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

64377 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)(5,7)(6,8)(9,10)(11,12,15,13)(14,16,18,17)(19,20)(21,25)(22,26)(23,24);(3,4,7,5)(6,8,10,9)(11,12,15,13)(14,16,18,17)(20,24)(22,25);(3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,17)(16,18)(19,21)(20,25)(22,24)(23,26);(4,5)(6,10)(12,13)(14,18)(19,22)(20,21)(23,25)(24,26);(1,2);(3,5,7,4)(6,9,10,8)(11,13,15,12)(14,16,18,17)(19,21)(20,25)(22,24)(23,26);(3,6)(4,8)(5,9)(7,10)(11,14)(12,16)(13,17)(15,18)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232068
maximal normal subgroup of	C₂×#72232068
maximal quotient of	C₂×#72232119

GAP

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

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

C2×#8126749 #72324815